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RD Chapter 6 Factorisation of Polynomials Ex 6.2 Solutions

Question - 1 : - If f(x) = 2x3 – 13x2 +17x + 12, find
(i) f (2)
(ii) f (-3)
(iii) f(0)

Answer - 1 : -

f(x) = 2x3 – 13x2 + 17x + 12
(i) f(2) = 2(2)3 – 13(2)2 + 17(2) + 12
= 2 x 8-13 x 4+17 x 2+12
= 16-52 + 34 + 12
= 62 – 52
= 10
(ii) f(-3) = 2(-3)3 – 13(-3)2 + 17 x (-3)+ 12
= 2 x (-27) – 13 x 9 + 17 x (-3) + 12
= -54 – 117 -51 + 12
= -222 + 12
= -210
(iii) f(0) = 2 x (0)3 – 13(0)2 + 17 x 0 + 12
= 0-0 + 0+ 12 = 12

Question - 2 : - Verify whether the indicated numbers are zeros of the polynomials corresponding to them in the following cases: [NCERT]

Answer - 2 : -


Question - 3 : - If x = 2 is a root of the polynomial f(x) = 2x2-3x+ la, find the value of a.

Answer - 3 : -

p(x) =2x2 – 3x + 7a
∵ x = 2is its zero, then
p(0) = 0
∴ p( 2)= 2(2)2 – 3×2 + la = 0
⇒2 x 4-3x2 + 7a = 0
⇒ 8 – 6+ 7o = 0
⇒2 + 7a= 0
⇒ 7a =-2 ⇒ a =−27
∴ Hencea = −27

Question - 4 : - If x = –12 is a zero of the polynomial p(x) = 8x3 –ax2 – x + 2, find the value of a.

Answer - 4 : -


Question - 5 : - If x = 0 and x = -1 are the roots of the polynomial f(x) = 2x3 – 3x2 + ax + b, find the value of a and b.

Answer - 5 : -

f(x) = 2x3 – 3x2 + ax + b
∵ x = 0and x = -1 are its zeros
∴ f(0) =0 and f(-1) = 0
Now, f(0) = 0
⇒ 2(0)3 – 3(0)2 + a x 0 + b = 0
⇒ 0-0 +0 + b= 0
∴ b = 0
and f(-1) = 0
⇒ 2(-1)3 –3(-1)2 + a(-1) + b = 0
⇒ 2 x (-1) – 3 x 1 + a x (-1) + b = 0
⇒ -2-3-a + b = 0
⇒ -2-3-a+ 0 = 0
⇒ -5- a= 0=>a =-5
Hence a = -5, b = 0

Question - 6 : - Find the integral roots of the polynomial f(x) = x3 + 6x2 + 11x + 6.

Answer - 6 : -

f(x) = x3 + 6x2 + 11x + 6
Construct = 6 = ±1, ±2, +3, ±6
If x = 1, then
f(1) = (1)3 + 6(1)2 + 11 x 1 + 6
= 1+ 6+11+ 6 = 24
∵ f(x) ≠ 0, +0
∴ x = 1is not its zero
Similarly, f(-1) = (-1)3 + 6(-1)2 + 11(-1) + 6
= -1 + 6 x 1-11+6
=-1+6-11+6
= 12-12 = 0
∴ x = -1 is its zero
f(-2) = (-2)3 + 6(-2)2 + 11 (-2) + 6
= -8 + 24 – 22 + 6
= -30 + 30 = 0
∴ x = -2is its zero
f(-3) = (-3)3 + 6(-3)2 + 11 (-3) + 6
= -27 + 54 – 33 + 6 = 60 – 60 = 0
∴ x = -3 is its zero
x = -1, -2, -3 are zeros of f(x)
Hence roots of f(x) are -1, -2, -3

Question - 7 : - Find the rational roots of the polynomial f(x) = 2x3 + x2 – 7x – 6.

Answer - 7 : -


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