Question

C) Find p and q such that 3 and - 1 are the zeros
of f(x) = x4 + px3 + qx2 + 12x - 9.​

Answer 1

Answer:

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Step-by-step explanation:

we have to find the values of p and q such that 3 and - 1 are zeroes of f(x) = x⁴ + px³ + qx² + 12x - 9

Solution : we know, if x = a is a zero of polynomial, f(x) then f(a) must be equal to zero. i.e., f(a) = 0.

similarly, 3 and -1 are zeroes of f(x).

so, f(3) = 0

⇒ 3⁴ + p(3)³ + q(3)² + 12(3) - 9 = 0

⇒81 + 27p + 9q + 36 - 9 = 0

⇒27p + 9q + 108 = 0

⇒3p + q + 12 = 0

⇒3p + q = - 12.........(1)

again, f(-1) = 0

⇒(-1)⁴ + p(-1)³ + q(-1)² + 12(-1) - 9 = 0

⇒1 - p + q - 12 - 9 = 0

⇒-p + q - 20 = 0

⇒-p + q = 20 ..........(2)

from equations (1) and (2) we get,

p = -8, q = 12