Find the values of m and n in the polynomial | vedclass

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(B) Let $f(x) = 2x^{3} + mx^{2} + nx - 14$.Since $(x-1)$ is a factor of $f(x)$,by the Factor Theorem,$f(1) = 0$.$2(1)^{3} + m(1)^{2} + n(1) - 14 = 0$$

Vedclass · Accepted Answer

$m=9, n=3$

Vedclass · Answer

(B) Let $f(x) = 2x^{3} + mx^{2} + nx - 14$.Since $(x-1)$ is a factor of $f(x)$,by the Factor Theorem,$f(1) = 0$.$2(1)^{3} + m(1)^{2} + n(1) - 14 = 0$$2 + m + n - 14 = 0 \Rightarrow m + n = 12$ $...(1)$Since $(x+2)$ is a factor of $f(x)$,by the Factor Theorem,$f(-2) = 0$.$2(-2)^{3} + m(-2)^{2} + n(-2) - 14 = 0$$-16 + 4m - 2n - 14 = 0 \Rightarrow 4m - 2n = 30 \Rightarrow 2m - n = 15$ $...(2)$Adding equations $(1)$ and $(2)$:$(m + n) + (2m - n) = 12 + 15$$3m = 27 \Rightarrow m = 9$.Substituting $m = 9$ into equation $(1)$:$9 + n = 12 \Rightarrow n = 3$.Thus,$m = 9$ and $n = 3$.

Find the values of $m$ and $n$ in the polynomial $2x^{3} + mx^{2} + nx - 14$,such that $(x-1)$ and $(x+2)$ are its factors.

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