Question

For which value of a and b are the zeros of q(x)=x³+2x²+a also the zeros of the polynomial p(x)=x⁵-x⁴- 4x³+3x²+3x+b ?

Answer 1

◢GIVEN,

$zeros \: of \: q(x) = {x}^{3} + 2 {x}^{2} + a \\ are \: also \: zeros \: of \: \\ p(x) = {x}^{5} - {x}^{4} + 3 {x}^{2} + 3x + b$

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● q(x) is the factor of p(x)

So,

● The REMAINDER obtain an dividing p(x) by q(x) is 0

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◢REMAINDER = 0

$= > - (1 + a) {x}^{2} + 3(a + 1)x + (b - 2a) = 0$

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◢THEN,

$= > - (1 + a) \: = 0 \: \: \: \: .....Eq _{1} \\ \\ = > 3(a + 1) \: = 0 \: \: \: .....Eq _{2} \\ \\ = > (b - 2a) \: = 0 \: \: \: \: .....Eq _{3}$

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● BY, Solving Equations

$= > a + 1 \: = 0 \\ \\ = > a = - 1$

=> a = -1 _____________[◢ANSWER]

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$= > b - 2a = 0 \\ \\ = > b - 2( - 1) = 0 \\ \\ = > b + 2 = 0 \\ \\ = > b = - 2$

=> b = -2 _____________[◢ANSWER]

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● THUS, THE VALUE OF "a" AND "b" are "-1" AND "-2" RESPECTIVELY.

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☆$BE \: \: BRAINLY$☆