If the polynomial p(x) is divided by x^{2}+4x | vedclass

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Question

(A) According to the division algorithm for polynomials, the dividend $p(x)$ is given by the formula: $p(x) = (	ext{divisor} 	imes 	ext{quotient})

Vedclass · Accepted Answer

$x^{4}+57x+15$

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(A) According to the division algorithm for polynomials, the dividend $p(x)$ is given by the formula: $p(x) = (	ext{divisor} 	imes 	ext{quotient}) + 	ext{remainder}$.Given:Divisor $d(x) = x^{2}+4x+2$Quotient $q(x) = x^{2}-4x+14$Remainder $r(x) = 9x-13$Substituting these values into the formula:$p(x) = (x^{2}+4x+2)(x^{2}-4x+14) + (9x-13)$First, multiply the divisor and the quotient:$(x^{2}+4x+2)(x^{2}-4x+14) = x^{2}(x^{2}-4x+14) + 4x(x^{2}-4x+14) + 2(x^{2}-4x+14)$$= (x^{4}-4x^{3}+14x^{2}) + (4x^{3}-16x^{2}+56x) + (2x^{2}-8x+28)$Combine like terms:$= x^{4} + (-4x^{3}+4x^{3}) + (14x^{2}-16x^{2}+2x^{2}) + (56x-8x) + 28$$= x^{4} + 0x^{3} + 0x^{2} + 48x + 28$Now, add the remainder $r(x) = 9x-13$:$p(x) = (x^{4}+48x+28) + (9x-13)$$p(x) = x^{4} + (48x+9x) + (28-13)$$p(x) = x^{4} + 57x + 15$.

If the polynomial $p(x)$ is divided by $x^{2}+4x+2$, then the quotient polynomial $x^{2}-4x+14$ and the remainder polynomial $9x-13$ are obtained. Find the polynomial $p(x)$.

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