How do you use the remainder theorem and the factor theorem to determine whether (b+4) is a factor of $(b^{3} + 3 b^{2} - b + 12)$ $(b^3 + 3b^2 − b + 12)$ ?

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Answer 1 · 2016-04-14T09:14:34

$(b + 4)$ $(b+4)$ is a factor of $b^{3} + 3 b^{2} - b + 12$ $b^3+3b^2-b+12$

Remainder theorem states that if we divide a polynomial function $f (x)$ $f(x)$ by $(x - a)$ $(x-a)$ , the remainder is $f (a)$ $f(a)$ .

Hence, if $(x - a)$ $(x-a)$ is a factor of $f (x)$ $f(x)$ , $f (a) = 0$ $f(a)=0$ (This is factor theorem.)

As we have to determine whether $(b + 4)$ $(b+4)$ is a factor of $f (b) = b^{3} + 3 b^{2} - b + 12$ $f(b)=b^3+3b^2-b+12$ or not,

we should evaluate $f (- 4)$ $f(-4)$ which is equal to

${(- 4)}^{3} + 3 {(- 4)}^{2} - (- 4) + 12 = - 64 + 48 + 4 + 12 = 0$ $(-4)^3+3(-4)^2-(-4)+12=-64+48+4+12=0$

Hence $(b + 4)$ $(b+4)$ is a factor of $b^{3} + 3 b^{2} - b + 12$ $b^3+3b^2-b+12$ .

How do you use the remainder theorem and the factor theorem to determine whether (b+4) is a factor of (b3+3b2−b+12)(b^3 + 3b^2 − b + 12)?