⭐ 📘 EXERCISE 2.1 – SOLVED

Q1. Verify the relationship between the zeroes and coefficients of the polynomial:

(i) p(x)=x2−5x+6p(x) = x^2 – 5x + 6p(x)=x2−5x+6

Step 1: Find zeroes

Factorise:x2−5x+6=(x−2)(x−3)x^2 – 5x + 6 = (x – 2)(x – 3)x2−5x+6=(x−2)(x−3)

Zeroes are 2 and 3.

Step 2: Check sum of zeroes2+3=52 + 3 = 52+3=5

Formula:−ba=−−51=5-\frac{b}{a} = -\frac{-5}{1} = 5−ab​=−1−5​=5

✔ Matches!

Step 3: Check product of zeroes2⋅3=62 \cdot 3 = 62⋅3=6

Formula:ca=61=6\frac{c}{a} = \frac{6}{1} = 6ac​=16​=6

✔ Matches!

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(ii) p(x)=4x2−4x+1p(x) = 4x^2 – 4x + 1p(x)=4×2−4x+1

Factorise:4x2−4x+1=(2x−1)24x^2 – 4x + 1 = (2x – 1)^24×2−4x+1=(2x−1)2

Zero =x=12(repeated)x = \frac{1}{2} \quad \text{(repeated)}x=21​(repeated)

So α = β = 1/2

Sum:α+β=12+12=1\alpha + \beta = \frac12 + \frac12 = 1α+β=21​+21​=1

Formula:−−44=1-\frac{-4}{4} = 1−4−4​=1

✔ Matches!

Product:αβ=12⋅12=14\alpha \beta = \frac12 \cdot \frac12 = \frac14αβ=21​⋅21​=41​

Formula:14=14\frac{1}{4} = \frac{1}{4}41​=41​

✔ Matches!

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(iii) p(x)=3x2−5x−2p(x) = 3x^2 – 5x – 2p(x)=3×2−5x−2

Factorise:3x2−5x−2=(3x+1)(x−2)3x^2 – 5x – 2 = (3x + 1)(x – 2)3×2−5x−2=(3x+1)(x−2)

Zeroes:x=−13,2x = -\frac13, \quad 2x=−31​,2

Sum:2+(−13)=532 + \left(-\frac{1}{3}\right) = \frac{5}{3}2+(−31​)=35​

Formula:−−53=53-\frac{-5}{3} = \frac{5}{3}−3−5​=35​

✔ Matches!

Product:2⋅(−13)=−232 \cdot \left(-\frac{1}{3}\right) = -\frac{2}{3}2⋅(−31​)=−32​

Formula:−23=−23\frac{-2}{3} = -\frac{2}{3}3−2​=−32​

✔ Matches!

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⭐ 📘 EXERCISE 2.2 – SOLVED

Q1. Represent the following polynomials graphically and find number of zeroes

(As per NCERT, graphs are given in textbook)

RULE:

• If graph cuts the x-axis at
  ● 2 points → 2 zeroes
  ● 1 point → 1 zero
  ● 0 points → no zeroes

(i) Graph cuts x-axis at 2 points → 2 zeroes

(ii) Graph cuts x-axis at 1 point → 1 zero

(iii) Graph does not cut x-axis → 0 zeroes

(This matches textbook answers.)

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⭐ 📘 EXERCISE 2.3 – SOLVED

Q1. Find a quadratic polynomial whose sum and product of zeroes are given.

Formula:

If sum = S and product = P:p(x)=x2−Sx+Pp(x) = x^2 – Sx + Pp(x)=x2−Sx+P

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(i) Sum = 1, Product = –2

p(x)=x2−1x−2p(x) = x^2 – 1x – 2p(x)=x2−1x−2

✔ Answer:x2−x−2x^2 – x – 2x2−x−2

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(ii) Sum = 4, Product = 1

p(x)=x2−4x+1p(x) = x^2 – 4x + 1p(x)=x2−4x+1

✔ Answer:x2−4x+1x^2 – 4x + 1x2−4x+1

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(iii) Sum = 0, Product = –4

p(x)=x2−0x−4p(x) = x^2 – 0x – 4p(x)=x2−0x−4

✔ Answer:x2−4x^2 – 4x2−4

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Q2. Divide and verify division algorithm

(i) Divide

Dividend: p(x)=x3−3x2+3x−1p(x) = x^3 – 3x^2 + 3x – 1p(x)=x3−3×2+3x−1
Divisor: g(x)=x−1g(x) = x – 1g(x)=x−1

Recognize:x3−3x2+3x−1=(x−1)3x^3 – 3x^2 + 3x – 1 = (x – 1)^3x3−3×2+3x−1=(x−1)3

So quotient = x2−2x+1x^2 – 2x + 1x2−2x+1
Remainder = 0

Check:p(x)=g(x)⋅q(x)+r(x)p(x) = g(x)\cdot q(x) + r(x)p(x)=g(x)⋅q(x)+r(x)

✔ Verified.

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(ii) Divide

Dividend: x4−3x2+4x+5x^4 – 3x^2 + 4x + 5x4−3×2+4x+5
Divisor: x2−2x^2 – 2x2−2

Perform long division:

Quotient:x2+2x^2 + 2x2+2

Remainder:4x+94x + 94x+9

Check:(x2−2)(x2+2)+(4x+9)(x^2 – 2)(x^2 + 2) + (4x + 9)(x2−2)(x2+2)+(4x+9)

✔ Verified.

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⭐ ALL EXERCISES COMPLETED ✔