Polynomials: Class 10 Maths Chapter 2 Hinglish mein

Mathematics · Hinglish notes, simulations aur practice · Official NCERT chapter PDF ↗

Published 12 Oct 2026 · Updated 12 Oct 2026

60 min
1 simulation
15 practice questions
15 MCQs

Is chapter mein aap kya seekhenge

Degree se polynomial ko linear, quadratic ya cubic naam do, aur har ek ka general form likho.
Polynomial ki value p(k) nikaalo aur tay karo ki k uska zero hai ya nahi.
y = p(x) ke graph se polynomial ke zeroes padho: yeh x ki woh values hain jahan curve x-axis se milta hai.
Samjhao ki quadratic polynomial ke zyada se zyada do zeroes kyun hote hain, cubic ke zyada se zyada teen aur degree n wale polynomial ke zyada se zyada n.
Factorise karke quadratic polynomial ke zeroes nikaalo, aur verify karo ki unka sum −b/a hai aur product c/a.
Zeroes ka sum aur product diya ho, to quadratic polynomial likho.
Cubic polynomial ke zeroes aur coefficients ke beech ke teen relations check karo.

NCERT Mathematics Class 10, Chapter 2, pp. 10–23 par based original companion text. Yeh NCERT se juda nahi hai aur NCERT ne ise endorse nahi kiya hai.

2.1 Introduction

Maan lo ek ball zameen se seedhi upar 20 m/s se nikalti hai. Air resistance ko ignore karo, aur lo $g \approx 10 m / s^{2} .$ Tab $t$ seconds baad uski height lagbhag $h (t) = 20 t - 5 t^{2}$ metres hoti hai. Kuch values: $h (1) = 15,$ $h (2) = 20$ aur $h (3) = 15.$ Ball wapas zameen par kab aati hai? Humein woh times chahiye jab height zero ho. Try karo $t = 4 :$ $h (4) = 80 - 80 = 0.$ Yaani ball 4 seconds baad zameen par girti hai (Fig. 2.1).

3/3 $h (t) = 0$ hota hai $t = 0$ (launch) par aur $t = 4$ (landing) par.

Fig. 2.1Height ko zero banane wale inputs,

t = 0

aur

t = 4,

is polynomial ke zeroes hain:

20 t - 5 t^{2} .

Metres mein height aur seconds mein time ka graph, right mein values ki ek chhoti table ke saath. Pehle step mein curve ka sirf upar jaata aadha hissa bana hai, time 0 par height 0 se top point tak, yaani 2 seconds par 20 metres. Doosre step mein neeche aata aadha hissa dikhta hai, jo 4 seconds par height 0 par wapas aata hai. Teesre step mein time axis ke do points, 0 aur 4 seconds par, gol ghere mein hain aur table ki woh rows highlight hain jahan height 0 hai.

Numbers $0$ aur $4$ is polynomial ke liye khaas inputs hain, $20 t - 5 t^{2} :$ yeh uski value zero kar dete hain. Is chapter mein tum aise inputs dhoondhoge, unhe graph par dekhoge, aur unhe polynomial mein aane wale numbers se jodoge. Pehle Class 9 ka ek chhota sa recap.

Ek variable wale polynomial ki degree us variable ki sabse badi power hoti hai. Degree 1, 2 aur 3 wale polynomials ke apne naam hain: linear, quadratic aur cubic (Table 2.1). Har general form mein letters $a,$ $b,$ $c,$ $d$ real numbers hain jinhe coefficients kehte hain. Leading coefficient $a$ zero nahi hona chahiye, warna degree gir jaayegi.

Table 2.1Degree se naam diye gaye polynomials (har general form mein

a \neq 0

)

Degree aur naam	General form	Examples
1: linear	$a x + b$	$7 - 2 x,$ $\sqrt{3} y + 1$
2: quadratic	$a x^{2} + b x + c$	$x^{2} - 9,$ $5 u - u^{2}$
3: cubic	$a x^{3} + b x^{2} + c x + d$	$x^{3},$ $4 - 2 p^{3} + p$

Quadratic shabd Latin ke quadratus se aaya hai, jiska matlab hai 'square banaya hua': $x^{2}$ us square ka area hai jiski side hai $x .$ $\frac{1}{x} + 2$ aur $\sqrt{x} - 5$ jaise expressions polynomials nahi hain. Inmein $x$ denominator mein ya root sign ke andar hai, yaani uski power $- 1$ ya $\frac{1}{2}$ jaisi hai jo whole number nahi hai.

Polynomial ki value aur zero

Maan lo $p (x) = x^{2} + x - 6.$ $x$ ki jagah ek number $k$ rakho aur result nikaalo: yahi polynomial ki value hai jab input ho $k,$ aur ise likhte hain $p (k) .$ Jaise, $p (1) = 1 + 1 - 6 = - 4$ aur $p (0) = - 6.$ Dhyaan do ki $p (0)$ hamesha constant term hota hai. Ab try karo $x = 2$ aur $x = - 3 :$ $p (2) = 4 + 2 - 6 = 0$ aur $p (- 3) = 9 - 3 - 6 = 0.$

Definition

Polynomial ka zero

$p (x)$ ka zero koi bhi aisa real number $k$ hai jo polynomial ki value zero kar de, yaani $p (k) = 0.$

Example $2$ aur $- 3$ zeroes hain jab polynomial ho $x^{2} + x - 6,$ aur $4$ ek zero hai jab polynomial ho $20 t - 5 t^{2} .$

Linear polynomial ka zero nikaalna linear equation solve karne jaisa hi hai, jo tumne Class 9 mein kiya tha. Maan lo polynomial hai $5 x + 8,$ to zero $k$ ko satisfy karna hoga $5 k + 8 = 0,$ isliye $k = - \frac{8}{5} .$ General roop se, agar $a k + b = 0$ aur $a \neq 0,$ to:

Linear polynomial ka zero

k = - \frac{b}{a}

(2.1)

jahan

$b$: constant term
$a$: $x$ ka coefficient

Dhyaan do ki $- \frac{b}{a}$ mein sirf do coefficients hi use hote hain. Kya quadratic ke zeroes, jo do bhi ho sakte hain, $a,$ $b$ aur $c$ se predict kiye ja sakte hain? Is par hum Section 2.3 mein wapas aayenge. Pehle dekhte hain ki graph par zeroes kaise dikhte hain.

Quick check

4 mein se 1

Inmein se kaun sa quadratic polynomial hai?

x^{3} - x^{2}

4 - 3 x^{2} + x

x^{2} + \frac{1}{x}

2 x + 7

Answer dekho

Answer (B) $4 - 3 x^{2} + x$

4 - 3 x^{2} + x

ki degree 2 hai.

x^{3} - x^{2}

ki degree 3 hai, aur

x^{2} + \frac{1}{x}

polynomial hi nahi hai, kyunki

\frac{1}{x}

matlab

x

ki power

- 1.

2.2 Polynomial ke Zeroes ka Geometrical Meaning

Graph par zero kaisa dikhta hai? Fig. 2.1 mein zeroes ne bataya ki ball zameen par kab thi. Graph kisi bhi polynomial ke liye yahi cheez dikhata hai. Maan lo graph hai $y = p (x),$ to value $p (k)$ curve ki woh height hai jo $x$ -axis ke upar ya neeche hoti hai, wahan jahan $x = k .$ Zero woh input hai jahan yeh height bilkul 0 ho, isliye zero par curve $x$ -axis par hota hai. Chalo is idea ko linear, quadratic aur cubic polynomials par test karte hain.

Linear polynomials

Class 9 mein tumne graph banaya tha $y = a x + b,$ jo ek straight line hai. Maan lo $y = 4 - 2 x .$ Is par kuch points hain $(0, 4),$ $(1, 2),$ $(2, 0)$ aur $(3, - 2) .$ Line neeche ki taraf slope karti hai aur $x$ -axis ko $(2, 0)$ par cross karti hai (Fig. 2.2). Aur $2$ bilkul is polynomial ka zero hai, $4 - 2 x,$ kyunki $4 - 2 (2) = 0.$

3/3 $x$ -axis par $y = 0,$ isliye $4 - 2 x = 0$ aur $x = 2.$

Fig. 2.2

4 - 2 x

ka zero wahan hai jahan uski line

x

-axis ko cross karti hai.NCERT Fig. 2.1 · p. 12

Ek square grid jismein x minus 1 se 5 tak aur y minus 3 se 5 tak hai. Pehle step mein neeche girti line y barabar 4 minus 2x chaar marked points se hokar bani hai: (0, 4), (1, 2), (2, 0) aur (3, minus 2). Doosre step mein point (2, 0), jahan line x-axis ko cross karti hai, gol ghere mein hai aur label kiya gaya hai, saath mein note hai ki line x-axis se ek baar milti hai. Teesre step mein right mein working dikhti hai: x-axis par y 0 hai, isliye 4 minus 2x barabar 0 aur x barabar 2.

Yahi reasoning kisi bhi line $y = a x + b$ par chalti hai, jab $a \neq 0.$ $x$ -axis ke points par hota hai $y = 0,$ isliye crossing point ko satisfy karna hoga $a x + b = 0,$ jisse milta hai $x = - \frac{b}{a} .$ Jo line flat nahi hai woh $x$ -axis ko sirf ek baar kaatti hai, isliye har linear polynomial ka bilkul ek zero hota hai, crossing point par, yaani $(- \frac{b}{a}, 0) .$

Quadratic polynomials

Ab quadratic lo $p (x) = x^{2} + 2 x - 3.$ Table 2.2 mein whole numbers ke liye $y = x^{2} + 2 x - 3$ ki values di gayi hain; inki range hai $- 4$ se $2.$

Table 2.2

y = x^{2} + 2 x - 3

ki valuesNCERT Table 2.1 · p. 13

$x$	$- 4$	$- 3$	$- 2$	$- 1$	$0$	$1$	$2$
$y$	$5$	$0$	$- 3$	$- 4$	$- 3$	$0$	$5$

In saat points ko plot karo aur smooth curve se jodo: tumhe Fig. 2.3 wala U-shape milega. Table dikhati hai ki $y = 0$ hota hai jab $x = - 3$ aur jab $x = 1,$ isliye yahi zeroes hain. Graph par yeh bilkul wahi jagah hain jahan curve $x$ -axis se milta hai: $(- 3, 0)$ aur $(1, 0) .$

3/3Curve $x$ -axis se in points par milta hai: $(- 3, 0)$ aur $(1, 0) .$

Fig. 2.3

x^{2} + 2 x - 3

ke zeroes hain

- 3

aur

1 :

curve

x

-axis se thik wahin milta hai jahan yahi values leta hai

x .

NCERT Fig. 2.2 · p. 13

Ek grid jismein x minus 5 se 3 tak aur y minus 5 se 6 tak hai, right mein notes ke saath. Pehle step mein sirf table ke saat points plot hain, (minus 4, 5) se neeche (minus 1, minus 4) tak aur wapas upar (2, 5) tak. Doosre step mein ek smooth U-shaped curve unhe jodta hai. Teesre step mein woh do points jahan curve x-axis se milta hai, (minus 3, 0) aur (1, 0), gol ghere mein hain aur label kiye gaye hain, aur note kehta hai ki zeroes minus 3 aur 1 hain.

SimulationTable khud plot karo

Dono axes par minus 6 se 6 tak ek khaali square grid. Har tap ek labelled point jodta hai jo whole numbers par snap hota hai, aur grid ke paas ek list har point ke coordinates deti hai aur batati hai ki woh kahan hai, jaise x-axis par ya kisi quadrant mein. Points ko drag karke theek kiya ja sakta hai.

Fig. 2.4

y = x^{2} + 2 x - 3

ke points plot karne ke liye grid par tap karo aur U-shape banta dekho.

Try karo

Plot karo $(- 4, 5),$ $(- 3, 0),$ $(- 2, - 3),$ $(- 1, - 4),$ $(0, - 3),$ $(1, 0)$ aur $(2, 5) .$ List ke hisaab se kaun se do points $x$ -axis par hain?
Kaun sa plot kiya point sabse neeche hai? U-shape kis vertical line ke around symmetric hai, aur woh line dono zeroes se kaise judi hai?

Har quadratic polynomial ka graph isi tarah ka curve hota hai, jise parabola (परवलय) kehte hain. Woh kis taraf khulta hai, yeh $a x^{2} + b x + c$ mein $a$ ke sign par depend karta hai (Fig. 2.5). Jab $a > 0,$ parabola cup ki tarah upar ki taraf khulta hai. Jab $a < 0,$ woh ulte rakhe bowl ki tarah neeche ki taraf khulta hai. Ball ke liye $a = - 5,$ isiliye Fig. 2.1 mein uski height ka graph neeche ki taraf khulta hai.

Definition

Parabola

Woh curve jo $y = a x^{2} + b x + c$ ka graph hai, jab $a \neq 0.$ Yeh upar ki taraf khulta hai jab $a > 0$ aur neeche ki taraf jab $a < 0.$

Example $y = x^{2} + 2 x - 3$ ka graph upar ki taraf khulta hai, kyunki $a = 1.$

2/2 $y = 6 + x - x^{2}$ mein $a = - 1 < 0,$ isliye yeh neeche ki taraf khulta hai.

Fig. 2.5

a

ka sign tay karta hai ki parabola kis taraf khulega.

Ek grid jismein x minus 4 se 4 tak aur y minus 5 se 5 tak hai. Pehle step mein solid parabola y barabar x square minus 4 bana hai; yeh upar ki taraf khulta hai aur note kehta hai ki a 0 se bada ho to upar khulta hai. Doosre step mein dashed parabola y barabar 6 plus x minus x square dikhta hai; yeh neeche ki taraf khulta hai, note kehta hai ki a 0 se chhota ho to neeche khulta hai, aur pehla parabola halka pad jaata hai.

Do zeroes, ek zero ya koi nahi

Parabola aur $x$ -axis sirf teen tarah se mil sakte hain. Socho ki parabola $y = x^{2} - 2 x - 3$ upar khisak raha hai; aisa tab hota hai jab tum uska constant term badhate ho (Fig. 2.6).

Case (i): do crossings. Curve $x$ -axis ko do alag jagah par paar karta hai; inhe naam do $A$ aur $B .$ Har ek se ek zero milta hai, isliye do alag zeroes hote hain. $x^{2} - 2 x - 3$ ke liye yeh hain $- 1$ aur $3.$
Case (ii): sirf ek touch. Curve ka sabse neeche wala point $x$ -axis par tika hota hai. Socho ki $A$ aur $B$ ek doosre ki taraf khisakte hain jab tak mil nahi jaate. Tab quadratic ke do barabar zeroes hote hain, aur graph par humein ek zero dikhta hai. $(x - 1)^{2} = x^{2} - 2 x + 1$ ke liye woh zero hai $1.$
Case (iii): koi point nahi. Poora parabola $x$ -axis ke upar hota hai ( $a < 0$ hone par uske neeche). Koi zero nahi hota. Jaise, $x^{2} - 2 x + 3 = (x - 1)^{2} + 2$ kabhi bhi 2 se kam nahi hota.

1/3 $y = x^{2} - 2 x - 3$ axis ko do points par kaatta hai: $A (- 1, 0)$ aur $B (3, 0) .$

Fig. 2.6Quadratic polynomial ke do zeroes hote hain, do barabar zeroes, ya koi zero nahi.NCERT Figs. 2.3–2.5 · p. 14

Ek grid jismein x minus 3 se 5 tak aur y minus 5 se 6 tak hai, right mein case notes ke saath. Pehle step mein parabola y barabar x square minus 2x minus 3 x-axis ko A (minus 1, 0) aur B (3, 0) par kaatta hai, dono gol ghere mein hain: case (i), do zeroes. Doosre step mein wahi curve 4 units upar khisak gaya hai aur x-axis ko sirf (1, 0) par touch karta hai, jahan A barabar B likha hai: case (ii), do barabar zeroes. Teesre step mein woh 2 units aur upar khisak gaya hai aur poora x-axis ke upar hai: case (iii), koi zero nahi.

Neeche khulne wala parabola bhi yahi teen cases deta hai, bas ulte. Dono tarah, parabola $x$ -axis se kabhi teen points par nahi mil sakta. Isliye quadratic polynomial ke zyada se zyada do zeroes hote hain.

Zero explorer (Fig. 2.7) mein tum set karte ho $a,$ $b$ aur $c,$ aur graph aur uske zeroes turant update ho jaate hain. Teeno cases banane ki koshish karo, aur dekho ki $a$ shape ke saath kya karta hai.

InteractiveZero explorer

Minus 6 se 6 tak ke grid par ek interactive graph. Quadratic tab mein sliders aur plus-minus buttons y barabar a x square plus b x plus c mein a, b aur c ko 0.5 ke steps mein set karte hain. Parabola turant dobara banta hai, jahan bhi woh x-axis se milta hai us point par gol ghera aur label lagta hai, aur ek status box batata hai ki kitne zeroes hain, curve axis ko sirf touch karta hai ya nahi, aur woh kis taraf khulta hai. Example buttons do zeroes, touch karta parabola, koi zero nahi aur neeche khulta parabola load karte hain. Cubic tab a x cube plus b x square plus c x plus d ke liye yahi karta hai. Ek checkbox ek table kholta hai jo zeroes ke sum aur product ko minus b by a aur c by a se compare karti hai, aur cubic ke liye teeno cubic relations dikhati hai.

Fig. 2.7Coefficients badlo aur zeroes ko chalta dekho. Cubic tab aur coefficient check ka use Section 2.3 mein phir hoga.

Cubic polynomials

Cubic se hum kya expect karein? Maan lo $p (x) = x^{3} - 3 x^{2} - x + 3.$ $- 2$ se $4$ tak ke whole numbers ke liye iski values Table 2.3 mein hain.

Table 2.3

y = x^{3} - 3 x^{2} - x + 3

ki valuesNCERT Table 2.2 · p. 15

$x$	$- 2$	$- 1$	$0$	$1$	$2$	$3$	$4$
$y$	$- 15$	$0$	$3$	$0$	$- 3$	$0$	$15$

Value 0 hoti hai jab $x = - 1,$ $1$ aur $3.$ Terms ko group karne se yeh confirm hota hai: $p (x) = x^{2} (x - 3) - (x - 3),$ yaani $(x - 3) (x^{2} - 1),$ aur iske factors hain $(x - 3) (x - 1) (x + 1) .$

Fig. 2.8 ka graph upar chadhta hai, neeche mudta hai, phir upar mudta hai, aur $x$ -axis ko bilkul inhi teen points par cross karta hai.

Har cubic axis se teen baar nahi milta. Dekho $x^{3} - 2 x^{2} = x^{2} (x - 2),$ jismein factor $x^{2}$ ki wajah se $0$ ek repeated zero hai. Graph origin par axis ko sirf touch karta hai, aur uska crossing point hai $2,$ isliye zeroes hain $0$ aur $2.$ Ab dekho $x^{3} + x = x (x^{2} + 1),$ jismein factor $x^{2} + 1$ kabhi zero nahi hota (woh kam se kam 1 hai). Isliye $0$ hi akela zero hai, aur graph sirf ek baar cross karta hai.

1/3 $x^{3} - 3 x^{2} - x + 3$ axis ko yahan cross karta hai: $- 1,$ $1$ aur $3.$

Fig. 2.8Cubic polynomial ke kam se kam ek aur zyada se zyada teen zeroes hote hain.NCERT Figs. 2.6–2.8 · p. 16

Ek grid jismein x minus 2 se 4 tak aur y minus 6 se 6 tak hai, right mein notes ke saath. Pehle step mein solid curve y barabar x cube minus 3x square minus x plus 3 x-axis ko minus 1, 1 aur 3 par cross karta hai, teeno gol ghere mein. Doosre step mein woh halka pad jaata hai aur dashed curve y barabar x cube minus 2x square dikhta hai; yeh origin par axis ko touch karta hai aur 2 par cross karta hai, dono gol ghere mein. Teesre step mein dotted curve y barabar x cube plus x dikhta hai, jo sirf origin par cross karta hai.

Do baatein dhyaan do. Cubic ka graph axis se bahut neeche shuru hota hai aur bahut upar khatam hota hai (ya $a < 0$ hone par ulta), isliye use axis ko kam se kam ek baar cross karna hi padta hai. Aur woh zyada se zyada do baar mudta hai, isliye axis se teen baar se zyada nahi mil sakta.

PropertyZeroes aur degree

Degree $n$ wale polynomial ke $n$ se zyada zeroes nahi ho sakte: $y = p (x)$ ka graph $x$ -axis se $n$ baar ya usse kam milta hai.

Example 1NCERT Example 1 · p. 17

Graph se zeroes gino

Fig. 2.9 mein chaar curves hain, har ek kisi polynomial se is roop mein bana hai: $y = p (x) .$ Har ek ke liye $p (x)$ ke zeroes gino.

Fig. 2.9Example 1 ke graphs.NCERT Fig. 2.9 · p. 17

2 by 2 grid mein chaar chhote graphs, har ek mein x aur y axes hain. (a) Ek cubic curve neeche left se upar uthta hai, left mein x-axis ko ek baar cross karta hai, phir ek hump aur ek dip banata hai jo dono axis ke upar rehte hain. (b) Neeche khulne wala parabola jiska top bas x-axis ko touch karta hai. (c) Ek curve jo x-axis ko do points par cross karta hai aur phir right mein teesre point par bas touch karta hai. (d) Upar khulne wala parabola jo poora x-axis ke upar hai.

Pehle khud try karo. Phir solution ek-ek step karke kholo.

2.3 Polynomial ke Zeroes aur Coefficients ke beech Relationship

Section 2.1 ne humare saamne ek sawaal chhoda tha: kya quadratic ke zeroes uske coefficients par usi tarah depend karte hain jaise line ke liye $- \frac{b}{a}$ karta hai ((2.1))? Chalo experiment karte hain. Hum do quadratics ko factorise karenge, unke zeroes likhenge aur koi pattern dhoondhenge.

Pehle, $p (x) = 2 x^{2} + 2 x - 12.$ Har coefficient even hai, isliye 2 bahar nikaalo: $p (x) = 2 (x^{2} + x - 6),$ jiske factors hain $2 (x + 3) (x - 2) .$

Product sirf tab zero hota hai jab $x + 3 = 0$ ya $x - 2 = 0,$ isliye $p (x)$ ke zeroes hain $- 3$ aur $2.$

Ab, $q (x) = 4 x^{2} - 4 x - 3.$ Hum middle term split karte hain: aise do numbers dhoondho jinka product ho $a \times c = 4 \times (- 3) = - 12$ aur sum ho $b = - 4.$ Yeh numbers hain $- 6$ aur $2,$ isliye $- 4 x$ ban jaata hai $- 6 x + 2 x$ aur $q (x) = 4 x^{2} - 6 x + 2 x - 3.$ Pehle do terms aur aakhri do terms ko group karo: $2 x (2 x - 3) + 1 (2 x - 3),$ yaani $(2 x + 1) (2 x - 3) .$

To $q (x)$ ke zeroes hain $- \frac{1}{2}$ aur $\frac{3}{2} .$

Try karke dekho

Pattern pakdo

Pehle guess karo

q (x) = 4 x^{2} - 4 x - 3

ke liye zeroes ka sum hai

- \frac{1}{2} + \frac{3}{2} = 1.

Coefficients hain

a = 4,

b = - 4,

c = - 3

; inka kaun sa expression bhi 1 ke barabar hai?

\frac{b}{a}

- \frac{b}{a}

\frac{c}{a}

- \frac{c}{b}

Dono polynomials ke liye aakhri do columns bharo. Phir unhe sum aur product columns se compare karo.

Table 2.4Zeroes aur coefficients

Polynomial	Zeroes	Sum	Product	$- \frac{b}{a}$	$\frac{c}{a}$
$2 x^{2} + 2 x - 12$	$- 3,$ $2$	$- 1$	$- 6$
$4 x^{2} - 4 x - 3$	$- \frac{1}{2},$ $\frac{3}{2}$	$1$	$- \frac{3}{4}$

Tumhari entries apne aap save hoti hain.

Do examples sirf ittefaaq bhi ho sakte hain. Yeh raha kaaran ki yeh pattern har us quadratic par kyun chalta hai jiske zeroes hain.

DerivationQuadratic ke zeroes ka sum aur product

Maan lo quadratic hai $p (x) = a x^{2} + b x + c,$ jahan $a \neq 0,$ aur iske zeroes hain $α$ aur $β$ (Greek alpha aur beta). Factor Theorem se har zero ek factor deta hai: $x - α$ aur $x - β .$ Degree 2 hai, isliye bachta hai sirf ek constant multiplier $k :$ $p (x) = k (x - α) (x - β) .$

Brackets ko multiply karke kholne par milta hai $p (x) = k x^{2} - k (α + β) x + k α β .$

Do polynomials tabhi barabar hote hain jab unke matching coefficients barabar hon. $x^{2}$ terms, $x$ terms aur constants ko match karte hain:

\begin{aligned} a & = k \\ b & = - k (α + β) \\ c & = k α β \end{aligned}

Kyunki $k = a,$ $a$ se divide karne par (jo zero nahi hai) do relations milte hain.

Zeroes ka sum

α + β = - \frac{b}{a}

(2.2)

Zeroes ka product

α β = \frac{c}{a}

(2.3)

Shabdon mein: zeroes ka sum hai minus $x$ ka coefficient, divided by us term ka coefficient jismein hai $x^{2} .$ Zeroes ka product hai constant term, divided by usi term ka coefficient jismein hai $x^{2} .$

Zero explorer (Fig. 2.7) mein Compare the zeroes with the coefficients par tick karo aur apni pasand ke kisi bhi quadratic par relations test karo.

Example 2NCERT Example 2 · p. 20

Pehle zeroes, phir check

Is polynomial ke zeroes nikaalo: $5 t - 3 + 2 t^{2} .$ Phir check karo ki unka sum aur product coefficients se match karte hain.

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Example 3NCERT Example 3 · p. 20

Surd wale zeroes

Is polynomial ke zeroes nikaalo: $2 x^{2} - 14.$ Phir check karo ki unka sum aur product coefficients se match karte hain.

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Yeh relations ulti taraf se bhi kaam karte hain. Maan lo tumhe zeroes ka sum $S$ aur product $P$ pata hai. Agar hum lein $a = 1,$ to relations dete hain $b = - S$ aur $c = P,$ isliye in zeroes wala ek quadratic hai

Zeroes ke sum aur product se quadratic

x^{2} - S x + P

(2.4)

Kisi bhi non-zero constant $k$ se multiply karne par zeroes nahi badalte. Isliye in zeroes wale har quadratic ka form hota hai $k (x^{2} - S x + P) .$ Fig. 2.11 mein $k$ badalne se parabola khinchta ya ulta ho jaata hai, lekin woh phir bhi $x$ -axis ke unhi do points se guzarta hai.

3/3 $y = - x^{2} + x + 2$ neeche ki taraf khulta hai, zeroes wahi hain.

Fig. 2.11Same zeroes, kai quadratics:

k (x^{2} - x - 2)

ke zeroes

- 1

aur

2

hain, jab tak

k \neq 0.

Ek grid jismein x minus 3 se 4 tak aur y minus 5 se 6 tak hai, aur do points (minus 1, 0) aur (2, 0) poore samay gol ghere mein hain. Pehle step mein k barabar 1 ke liye solid parabola y barabar x square minus x minus 2 dono gheron se guzarta hai. Doosre step mein k barabar 2 ka dashed parabola dikhta hai; yeh zyada patla hai aur do guna gehra jaata hai, unhi gheron se hokar. Teesre step mein k barabar minus 1 ka dotted parabola dikhta hai; yeh neeche ki taraf khulta hai, aur phir se unhi do gheron se guzarta hai.

Example 4NCERT Example 4 · p. 21

Zeroes ke sum aur product se quadratic banao

Ek quadratic ke zeroes ka sum $\frac{7}{2}$ hai aur product $\frac{3}{2} .$ Whole-number coefficients wala aisa ek quadratic likho.

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Example

Board-exam style: bina solve kiye relations use karo

Agar polynomial hai $x^{2} - 6 x + 4,$ aur uske zeroes hain $α$ aur $β$ , to nikaalo (a) $α^{2} + β^{2}$ aur (b) $\frac{1}{α} + \frac{1}{β},$ aur woh bhi bina nikaale $α$ aur $β .$

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Quick check

4 mein se 1

3 x^{2} - 6 x + 2

ke zeroes ka sum kya hai?

2

- 2

\frac{2}{3}

- 6

Answer dekho

Answer (A) $2$

- \frac{b}{a} = - \frac{- 6}{3} = 2.

Minus sign bhoolne par aata hai

- 2,

aur

\frac{2}{3}

to hai

\frac{c}{a},

yaani product.

Cubic ke zeroes aur coefficients

Kya cubic bhi aisa hi koi rule follow karta hai? Maan lo $p (x) = 2 x^{3} + 3 x^{2} - 8 x + 3.$ Substitute karne se pata chalta hai ki $1,$ $- 3$ aur $\frac{1}{2}$ zeroes hain:

$p (1) = 2 + 3 - 8 + 3 = 0$
$p (- 3) = - 54 + 27 + 24 + 3 = 0$
$p (\frac{1}{2}) = \frac{1}{4} + \frac{3}{4} - 4 + 3 = 0$

Chauthe zero ki koi jagah nahi hai, kyunki cubic ke zyada se zyada teen zeroes hote hain. Table 2.5 inke combinations ko coefficients se compare karti hai: $a = 2,$ $b = 3,$ $c = - 8$ aur $d = 3.$

Table 2.5

2 x^{3} + 3 x^{2} - 8 x + 3

ke zeroes

1,

- 3,

\frac{1}{2}

Combination	Zeroes se	Coefficients se
zeroes ka sum	$1 - 3 + \frac{1}{2} = - \frac{3}{2}$	$- \frac{b}{a} = - \frac{3}{2}$
pairs mein products ka sum	$- 3 - \frac{3}{2} + \frac{1}{2} = - 4$	$\frac{c}{a} = \frac{- 8}{2} = - 4$
zeroes ka product	$1 \times (- 3) \times \frac{1}{2} = - \frac{3}{2}$	$- \frac{d}{a} = - \frac{3}{2}$

To cubic ke teen relations hote hain, do nahi. Beech wala naya hai: zeroes ko pairs mein multiply karo, $α β,$ $β γ$ aur $γ α,$ aur teeno results jodo. Yeh raha kaaran ki yeh relations har cubic par kyun chalte hain.

DerivationCubic ke relations

Maan lo $α,$ $β,$ $γ$ (gamma) zeroes hain is cubic ke: $p (x) = a x^{3} + b x^{2} + c x + d .$ Pehle ki tarah, Factor Theorem se milta hai $p (x) = a (x - α) (x - β) (x - γ) .$

Brackets ko multiply karke kholo aur terms ko har power ke hisaab se ikattha karo, jahan variable hai $x .$ Product $(x - α) (x - β) (x - γ)$ ka result hai $x^{3} - (α + β + γ) x^{2} + (α β + β γ + γ α) x - α β γ .$

$a$ se multiply karo aur isse compare karo: $a x^{3} + b x^{2} + c x + d .$ Tab $b = - a (α + β + γ),$ $c = a (α β + β γ + γ α)$ aur $d = - a α β γ .$ Har ek ko $a$ se divide karne par neeche ke teen relations milte hain.

Cubic ke zeroes aur coefficients

\begin{aligned} α + β + γ & = - \frac{b}{a} \\ α β + β γ + γ α & = \frac{c}{a} \\ α β γ & = - \frac{d}{a} \end{aligned}

(2.5)

Signs baari-baari badalte hain, is order mein: $b$ , $c$ , $d :$ minus, plus, minus.

Example 5NCERT Example 5 · p. 22

Cubic ke zeroes check karo

Substitution se dikhao ki $2,$ $- 1$ aur $- \frac{2}{3}$ zeroes hain is polynomial ke: $p (x) = 3 x^{3} - x^{2} - 8 x - 4.$ Phir iske liye teeno zero–coefficient relations check karo.

NCERT ne Example 5 ko exam ke bahar mark kiya hai. Phir bhi cubic relations ki practice ke liye yeh sabse badhiya hai.

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Quick check

3 mein se 1

x^{3} - 4 x^{2} + x + 6

ke zeroes ka product kya hai?

Answer dekho

Answer -6

- \frac{d}{a} = - \frac{6}{1} = - 6.

Zeroes hain

- 1,

2

aur

3,

aur

(- 1) (2) (3) = - 6.

Agar use karo

\frac{d}{a},

jaise quadratic ke product mein, to sign galat aata hai.

Practice · 10 questions

Practice questions answers ke saath (NCERT “Practice 2.2”)

1
Short answerEasyNCERT Exercise 2.2, Q1
Har quadratic polynomial ko factorise karke uske zeroes nikaalo. Phir check karo ki unka sum aur product coefficients se match karte hain.
1. $x^{2} - 3 x - 28$
  Solution dikhao
  Answer $7$ aur $- 4$
  Product $- 28$ aur sum $- 3$ wale do numbers hain $- 7$ aur $4.$ Isliye polynomial hai $(x - 7) (x + 4),$ aur zeroes hain $7$ aur $- 4.$
  Sum: $3 = - \frac{- 3}{1} .$ Product: $- 28 = \frac{- 28}{1} .$
2. $9 y^{2} - 12 y + 4$
  Solution dikhao
  Answer $\frac{2}{3}$ aur $\frac{2}{3}$ (barabar zeroes)
  Yeh ek perfect square hai, $(3 y - 2)^{2} .$ Dono zeroes hain $\frac{2}{3} .$
  Sum: $\frac{4}{3} = - \frac{- 12}{9} .$ Product: $\frac{4}{9} = \frac{4}{9} .$
3. $7 x - 12 + 10 x^{2}$
  Solution dikhao
  Answer $\frac{4}{5}$ aur $- \frac{3}{2}$
  Standard form: $10 x^{2} + 7 x - 12.$ Yahan $a c = - 120;$ $15$ aur $- 8$ ka product $- 120$ hai aur sum $7.$
  Grouping se milta hai $(5 x - 4) (2 x + 3),$ isliye zeroes hain $\frac{4}{5}$ aur $- \frac{3}{2} .$
  Sum: $- \frac{7}{10} = - \frac{b}{a} .$ Product: $- \frac{6}{5} = \frac{- 12}{10} .$
4. $3 v^{2} - 12 v$
  Solution dikhao
  Answer $0$ aur $4$
  Common factor $3 v$ bahar nikaalo, to milega $3 v (v - 4) .$ Zeroes hain $0$ aur $4.$
  Yahan $c = 0.$ Sum: $4 = - \frac{- 12}{3} .$ Product: $0 = \frac{0}{3} .$
2
Short answerMediumNCERT Exercise 2.2, Q1
Zeroes nikaalo, phir unke sum aur product ko coefficients se check karo.
1. $p^{2} - 11$
  Solution dikhao
  Answer $\sqrt{11}$ aur $- \sqrt{11}$
  $11$ ko $(\sqrt{11})^{2}$ likho aur difference of squares use karo. Zeroes hain $\sqrt{11}$ aur $- \sqrt{11} .$
  Koi $p$ term nahi hai, isliye $b = 0.$ Sum: $0 = - \frac{0}{1} .$ Product: $- 11 = \frac{- 11}{1} .$
2. $5 x^{2} + 4 x - 1$
  Solution dikhao
  Answer $\frac{1}{5}$ aur $- 1$
  Yahan $a c = - 5;$ $5$ aur $- 1$ ka product $- 5$ hai aur sum hai $4.$ Isliye $5 x^{2} + 5 x - x - 1$ group hokar banta hai $(5 x - 1) (x + 1) .$
  Zeroes: $\frac{1}{5}$ aur $- 1.$ Sum: $- \frac{4}{5} = - \frac{b}{a} .$ Product: $- \frac{1}{5} = \frac{c}{a} .$
3
Short answerEasyNCERT Exercise 2.2, Q2
Aisa quadratic polynomial nikaalo jiske zeroes ka sum aur product diye gaye hain.
1. Sum $- 2,$ product $- 15$
  Solution dikhao
  Answer $x^{2} + 2 x - 15$
  $S = - 2$ aur $P = - 15,$ isliye $x^{2} - S x + P$ ban jaata hai $x^{2} + 2 x - 15.$
2. Sum $\frac{1}{3},$ product $- 2$
  Solution dikhao
  Answer $3 x^{2} - x - 6$
  $x^{2} - \frac{1}{3} x - 2.$ $3$ se multiply karo to milega $3 x^{2} - x - 6.$
3. Sum $0,$ product $- \frac{3}{4}$
  Solution dikhao
  Answer $4 x^{2} - 3$
  $x^{2} - 0 x - \frac{3}{4} .$ $4$ se multiply karo to milega $4 x^{2} - 3,$ jiske zeroes hain $\pm \frac{\sqrt{3}}{2} .$
4. Sum $2 \sqrt{3},$ product $3$
  Solution dikhao
  Answer $x^{2} - 2 \sqrt{3} x + 3$
  $x^{2} - 2 \sqrt{3} x + 3.$ Yeh hai $(x - \sqrt{3})^{2},$ isliye dono zeroes hain $\sqrt{3} .$
4
Short answerMediumNCERT Exercise 2.2, Q2
Aisa quadratic polynomial nikaalo jiske zeroes ka sum aur product diye gaye hain.
1. Sum $- \frac{3}{5},$ product $0$
  Solution dikhao
  Answer $5 x^{2} + 3 x$
  $x^{2} + \frac{3}{5} x + 0.$ $5$ se multiply karo to milega $5 x^{2} + 3 x,$ jiske zeroes hain $0$ aur $- \frac{3}{5} .$
2. Sum $- \frac{1}{2},$ product $- \frac{1}{2}$
  Solution dikhao
  Answer $2 x^{2} + x - 1$
  $x^{2} + \frac{1}{2} x - \frac{1}{2} .$ $2$ se multiply karo to milega $2 x^{2} + x - 1,$ jo hai $(2 x - 1) (x + 1) .$
5
Short answerMedium
Agar polynomial hai $3 x^{2} - 6 x + 1,$ aur uske zeroes hain $α$ aur $β$ , to har value nikaalo, bina nikaale $α$ aur $β .$
1. $\frac{1}{α} + \frac{1}{β}$
  Answer dekho
  Answer 6
  $α + β = 2$ aur $α β = \frac{1}{3} .$ Isliye $\frac{α + β}{α β} = 2 \div \frac{1}{3} = 6.$
2. $α^{2} β + α β^{2}$
  Answer dekho
  Answer 0.6667
  Factorise karo: $α β (α + β) = \frac{1}{3} \times 2 = \frac{2}{3} .$
3. $(α - β)^{2}$
  Answer dekho
  Answer 2.6667
  $(α - β)^{2}$ ko likh sakte hain $(α + β)^{2} - 4 α β,$ jo hai $4 - \frac{4}{3} = \frac{8}{3} .$
Quick answerHard
$x^{2} - 9 x + k$ ka ek zero doosre ka do guna hai. Nikaalo $k .$
Answer dekho
Answer 18
Sum hai $m + 2 m = 3 m = 9,$ isliye $m = 3$ aur zeroes hain $3$ aur $6.$ Phir $k$ product hai, yaani $18.$
Quick answerMedium
Bina solve kiye, $x^{2} + 17 x + 41$ ke zeroes ke baare mein tum kya keh sakte ho? (Iske do zeroes zaroor hain.)
dono positivedono negativeek positive, ek negativedono barabar
Answer dekho
Answer (B) dono negative
Product $41$ positive hai, isliye zeroes ka sign same hai. Unka sum $- 17$ negative hai, isliye dono negative hain. (Yeh lagbhag $- 2.9$ aur $- 14.1$ hain.)
8
Short answerMedium
Cubic polynomial lo $2 x^{3} - 7 x^{2} + 2 x + 3 :$
1. Iske zeroes ka sum nikaalo.
  Answer dekho
  Answer 3.5
  $- \frac{b}{a} = - \frac{- 7}{2} = \frac{7}{2} .$
2. Iske zeroes ko do-do lekar bane products ka sum nikaalo.
  Answer dekho
  Answer 1
  $\frac{c}{a} = \frac{2}{2} = 1.$
3. Iske zeroes ka product nikaalo.
  Answer dekho
  Answer -1.5
  $- \frac{d}{a} = - \frac{3}{2} .$
4. Iske do zeroes hain $3$ aur $1.$ (a) ka use karke teesra nikaalo.
  Answer dekho
  Answer -0.5
  $3 + 1 + γ = \frac{7}{2},$ isliye $γ = - \frac{1}{2} .$ (c) se check karo: $3 \times 1 \times (- \frac{1}{2}) = - \frac{3}{2} .$
Quick answerMedium
Assertion (A): $x^{2} + 6$ ka koi zero nahi hai. Reason (R): $y = x^{2} + 6$ ka graph upar ki taraf khulne wala parabola hai, jiska sabse neeche wala point hai $(0, 6) .$
A aur R dono sahi hain, aur R, A ko samjhata hai.A aur R dono sahi hain, lekin R, A ko nahi samjhata.A sahi hai lekin R galat hai.A galat hai lekin R sahi hai.
Answer dekho
Answer (A) A aur R dono sahi hain, aur R, A ko samjhata hai.
Dono sahi hain, kyunki $x^{2} + 6 \geq 6$ , chahe koi bhi ho $x .$ Parabola ka sabse neeche wala point $x$ -axis ke upar hai, isliye graph axis se kabhi nahi milta. Isi wajah se koi zero nahi hai.
Long answerHard
Aisa quadratic polynomial nikaalo jiske zeroes hain $2 + \sqrt{3}$ aur $2 - \sqrt{3} .$ Phir apna answer check karo; uski value nikaalo yahan: $2 + \sqrt{3} .$
Solution dikhao
Answer $x^{2} - 4 x + 1$
Sum: $\sqrt{3}$ wale terms cancel ho jaate hain, isliye $S = 4.$
Product: use karo $m^{2} - n^{2},$ to $P = 4 - 3 = 1.$
Isliye polynomial $x^{2} - 4 x + 1$ hai (ya iska koi bhi non-zero multiple).
Check: jab $x = 2 + \sqrt{3},$ tab $x^{2} = 7 + 4 \sqrt{3}$ aur $4 x = 8 + 4 \sqrt{3} .$ Isliye value hai $7 - 8 + 1 = 0.$

2.4 Summary

Fig. 2.12 poore chapter ko ek polynomial par dikhata hai: uska formula, uska graph, aur uske zeroes aur coefficients ke beech ka link.

3/3Sum $- 2 + 3 = 1 = - \frac{b}{a};$ product $(- 2) (3) = - 6 = \frac{c}{a} .$

Fig. 2.12Ek polynomial, teen views: formula, graph aur zero–coefficient relations.

Left mein ek grid par upar khulne wala parabola y barabar x square minus x minus 6 hai. Pehle step mein right side par formula aur uske coefficients dikhte hain: a barabar 1, b barabar minus 1, c barabar minus 6. Doosre step mein woh points jahan parabola x-axis se milta hai, minus 2 aur 3, gol ghere mein hain aur unke naam likhe hain. Teesre step mein ek shaded box dikhata hai ki minus 2 plus 3 barabar 1 hai, jo minus b by a hai, aur minus 2 guna 3 barabar minus 6 hai, jo c by a hai.

Recap

Summary aur key points

Linear, quadratic aur cubic degree 1, 2 aur 3 ke naam hain. Quadratic $a x^{2} + b x + c$ hota hai aur cubic $a x^{3} + b x^{2} + c x + d,$ jahan $a \neq 0$ (Section 2.1).
$k$ $p (x)$ ka zero hai jab $p (k) = 0.$ $a x + b$ ka zero $- \frac{b}{a}$ hai (Polynomial ki value aur zero).
$p (x)$ ka har zero us point ki $x$ -value hai jahan $y = p (x)$ ka graph $x$ -axis ko cross ya touch karta hai (Section 2.2).
Quadratic ka graph ek parabola hota hai. Yeh $a > 0$ hone par upar aur $a < 0$ hone par neeche khulta hai (Quadratic polynomials).
Quadratic ke do zeroes, do barabar zeroes ya koi zero nahi hota (Do zeroes, ek zero ya koi nahi). Cubic ke ek, do ya teen hote hain (Cubic polynomials).
Degree $n$ mein $n$ se zyada zeroes nahi ho sakte (Property).
Quadratic jiske zeroes hain $α$ aur $β :$ $α + β = - \frac{b}{a}$ aur $α β = \frac{c}{a}$ (Derivation).
Jis quadratic ke zeroes ka sum $S$ aur product $P$ ho, woh hai $k (x^{2} - S x + P),$ jahan $k \neq 0$ ((2.4)).
Cubic jiske zeroes hain $α,$ $β,$ $γ :$ sum hai $- \frac{b}{a},$ pairs mein products ka sum hai $\frac{c}{a}$ aur product hai $- \frac{d}{a}$ (Cubic ke zeroes aur coefficients).

Flash cards · 8

Formula yaad karo

1 / 8

Apna NCERT exercise question dhoondo

Hamare practice questions ncert.ai ke liye likhe gaye hain aur NCERT exercises ko follow karte hain. Apni textbook ke question se match karne wala question dhoondo; uska answer uske saath hai.

Tumhari NCERT textbook mein	Yahan practice question
NCERT Exercise 2.1, Q1	Practice 2.1 · Q1
NCERT Exercise 2.2, Q1	Practice 2.2 · Q1
NCERT Exercise 2.2, Q1	Practice 2.2 · Q2
NCERT Exercise 2.2, Q2	Practice 2.2 · Q3
NCERT Exercise 2.2, Q2	Practice 2.2 · Q4

Quick check: MCQs

Polynomial ki value aur zero4 questions
Cubic polynomials4 questions
2.3 Polynomial ke Zeroes aur Coefficients ke beech Relationship4 questions
Cubic ke zeroes aur coefficients3 questions

Glossary · 8

Key terms (English + Hindi)

coefficientगुणांकgunaank: Woh number jo kisi term mein variable ki power se multiply hota hai, jaise $- 4 x^{2}$ mein $- 4$ .Chapter mein pehli baar yahan aaya →
cubic polynomialत्रिघात बहुपदtrighaat bahupad: Degree 3 ka polynomial, $a x^{3} + b x^{2} + c x + d$ form ka, jahan $a \neq 0$ .Chapter mein pehli baar yahan aaya →

degreeघातghaat: Polynomial mein variable ki sabse badi power, jaise $2 x^{3} - x + 5$ ke liye 3.Chapter mein pehli baar yahan aaya →

Factor Theoremगुणनखंड प्रमेयgunankhand prameya: Agar $p (α) = 0$ , to $x - α$ $p (x)$ ka ek factor hai, aur ulta bhi sahi hai.Chapter mein pehli baar yahan aaya →

linear polynomialरैखिक बहुपदraikhik bahupad: Degree 1 ka polynomial, $a x + b$ form ka, jahan $a \neq 0$ .Chapter mein pehli baar yahan aaya →

parabolaपरवलयparvalay: Quadratic polynomial ka U-shaped graph. Yeh $a > 0$ hone par upar ki taraf aur $a < 0$ hone par neeche ki taraf khulta hai.Chapter mein pehli baar yahan aaya →
polynomial ka zeroबहुपद का शून्यकzero of a polynomialbahupad ka shoonyak: Aisa real number $k$ jiske liye $p (k) = 0$ . $y = p (x)$ ke graph par yeh woh jagah hai jahan curve $x$ -axis se milta hai.

quadratic polynomialद्विघात बहुपदdvighaat bahupad: Degree 2 ka polynomial, $a x^{2} + b x + c$ form ka, jahan $a \neq 0$ .Chapter mein pehli baar yahan aaya →

Polynomials: Class 10 Maths Chapter 2 Hinglish mein

Is chapter mein aap kya seekhenge

2.1 Introduction

Polynomial ki value aur zero

2.2 Polynomial ke Zeroes ka Geometrical Meaning

Linear polynomials

Quadratic polynomials

Do zeroes, ek zero ya koi nahi

Cubic polynomials

Practice 2.1: answers ke saath

2.3 Polynomial ke Zeroes aur Coefficients ke beech Relationship

Cubic ke zeroes aur coefficients

Practice questions answers ke saath (NCERT “Practice 2.2”)

2.4 Summary

Summary aur key points

Apna NCERT exercise question dhoondo

Quick check: MCQs

Key terms (English + Hindi)

Tum chapter ke end tak pahunch gaye