TOPICS

Exercise - 2.3

Polynomials

**Question-1 :-** Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following :

(i) p(x) = x³ – 3x² + 5x – 3, g(x) = x² – 2

(ii) p(x) = x⁴ – 3x² + 4x + 5, g(x) = x² + 1 – x

(iii) p(x) = x⁴ – 5x + 6, g(x) = 2 – x²

(i) p(x) = x³ – 3x² + 5x – 3, g(x) = x² – 2 q(x) = x - 3, r(x) = 7x - 9 Verification : p(x) = g(x) x q(x) + r(x) = (x² – 2)(x - 3) + 7x - 9 = x³ – 3x² - 2x + 6 + 7x – 9 = x³ – 3x² + 5x – 3

(ii) p(x) = x⁴ – 3x² + 4x + 5, g(x) = x² + 1 – x q(x) = x² + x - 3, r(x) = 8 Verification : p(x) = g(x) x q(x) + r(x) = (x² + 1 – x )(x² + x - 3) + 8 = x⁴ + x³ – 3x² + x² + x - 3 - x³ - x² + 3x + 8 = x⁴ – 3x² + 4x + 5

(iii) p(x) = x⁴ – 5x + 6, g(x) = 2 – x² q(x) = -2 – x², r(x) = -5x + 10 Verification : p(x) = g(x) x q(x) + r(x) = (2 – x²)(-2 – x²) + (-5x + 10) = -(4 - x⁴) - 5x + 10 = x⁴ – 5x + 6

**Question-2 :-** Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial :

(i) t² – 3, 2t⁴ + 3t³ – 2t² – 9t – 12

(ii) x² + 3x + 1, 3x⁴ + 5x³ – 7x² + 2x + 2

(iii) x³ – 3x + 1, x⁵ – 4x³ + x² + 3x + 1

(i) t² – 3, 2t⁴ + 3t³ – 2t² – 9t – 12 The remainder is 0. So, t² – 3 is a factor of 2t⁴ + 3t³ – 2t² – 9t – 12.

(ii) x² + 3x + 1, 3x⁴ + 5x³ – 7x² + 2x + 2 The remainder is 0. So, x² + 3x + 1 is a factor of 3x⁴ + 5x³ – 7x² + 2x + 2 .

(iii) x³ – 3x + 1, x⁵ – 4x³ + x² + 3x + 1 The remainder is not 0. So, x³ – 3x + 1 is not a factor of x⁵ – 4x³ + x² + 3x + 1 .

**Question-3 :-** Obtain all other zeroes of 3x⁴ + 6x³ – 2x² – 10x – 5, if two of its zeroes are √5/3 and -√5/3.

p(x) = 3x⁴ + 6x³ – 2x² – 10x – 5 Given that two zeroes are √5/3 and -√5/3. So, (x - √5/3)(x + √5/3) = x² - 5/3 [a² - b² = (a + b)(a - b)] q(x) = 3x² + 6x + 3 Now, p(x) = 3x⁴ + 6x³ – 2x² – 10x – 5 = (x - √5/3)(x + √5/3)(3x² + 6x + 3) = (x - √5/3)(x + √5/3) x 3 x (x² + 2x + 1) = (x - √5/3)(x + √5/3) x 3 x (x + 1)² = (x - √5/3)(x + √5/3) x 3 x (x + 1)(x + 1) x + 1 = 0 x = -1 Therefore, all zeroes are √5/3, -√5/3, -1 and -1.

**Question-4 :-** On dividing x³ – 3x² + x + 2 by a polynomial g(x), the quotient and remainder were x – 2 and –2x + 4, respectively. Find g(x).

p(x) = x³ – 3x² + x + 2 q(x) = x - 2 r(x) = -2x + 4 p(x) = g(x) x q(x) + r(x) x³ – 3x² + x + 2 = g(x) x (x - 2) + (-2x + 4) x³ – 3x² + x + 2 + 2x - 4 = g(x) x (x - 2) g(x) = (x³ – 3x² + 3x - 2)/(x - 2) g(x) = x² - x + 1

**Question-5 :-** Give examples of polynomials p(x), g(x), q(x) and r(x), which satisfy the division algorithm and

(i) deg p(x) = deg q(x)

(ii) deg q(x) = deg r(x)

(iii) deg r(x) = 0

(i) deg p(x) = deg q(x) We take an example in which deg p(x) = deg q(x) = 2

(ii) deg q(x) = deg r(x) We take an example in which deg p(x) = deg r(x) = 2

(iii) deg r(x) = 0 We take an example in which deg deg r(x) = 0

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