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  • For which values of a and b, are the zeroes of q(x) = x^3 + 2x^2 + a also the zeroes of the polynomial p(x) = x^5 – x^4 – 4x3 + 3x^2 + 3x + b? Which zeroes of p(x) are not the zeroes of q(x)?

For which values of a and b, are the zeroes of q(x) = x^3 + 2x^2 + a also the zeroes of the polynomial p(x) = x^5 - x^4 - 4x^3 + 3x^2 + 3x + b? Which zeroes of p(x) are not the zeroes of q(x)?

Answers (1)

Answer. [2, 1]

Solution.        

Given :-

q(x) = x^3 + 2x^2 + a, p(x) = x^5 - x^4 - 4x^3 + 3x^2 + 3x + b

Hence, q(x) is a factor of p(x) use divided algorithm

Since x^3 + 2x^2 + a is a factor hence remainder = 0

x^2(-1 - a) + 3x(1 + a) + (b - 2a) = 0.x^2 + 0.x + 0

By comparing

– 1 – a = 0                               b – 2a = 0

a = – 1                                     2a = b

                                                put a = –1

                                                2(–1) = b

                                                 b = –2

Hence, a = –1, b = –2

x^5 - x^4 - 4x^3 + 3x^2 + 3x - 2 = (x^3 + 2x^2 - 1) (x^2 - 3x + 2)

= (x^3 + 2x^2- 1) (x^2 - 2x - x + 2)

= (x^3 + 2x^2 - 1) (x(x - 2) - 1(x - 2))

= (x^3 + 2x^2 -1) (x - 2) (x - 1)

x – 2 = 0                                  x – 1 = 0

x = 2                                        x = 1

Hence, 2, 1 are the zeroes of p(x) which are not the zeroes of q(x).

 

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