How do you use the factor theorem to determine whether x-4 is a factor of $x^{3} - 21 x + 20$ $x^3-21x+20$ ?

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How do you use the factor theorem to determine whether x-4 is a factor of $x^{3} - 21 x + 20$ $x^3-21x+20$ ?

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Answer 1 · 2015-11-16T08:29:50

The factor theorem states that if $f (x_{0}) = 0$ $f(x_0)=0$ , then $(x - x_{0})$ $(x-x_0)$ divides $f (x)$ $f(x)$ . So, $(x - 4)$ $(x-4)$ divides $x^{3} - 21 x + 20$ $x^3-21x+20$ if and only if the polynomial is zero when $x = 4$ $x=4$ .

Let's do the computation:

$f (4) = 4^{3} - 21 \cdot 4 + 20 = 64 - 84 + 20 = 0$ $f(4)= 4^3 - 21*4+20 = 64-84+20 = 0$

So yes, $(x - 4)$ $(x-4)$ is a factor of $x^{3} - 21 x + 20$ $x^3-21x+20$ . Indeed, the three roots are $- 5$ $-5$ , $1$ $1$ and $4$ $4$ , so we have

$x^{3} - 21 x + 20 = (x - 1) (x - 4) (x + 5)$ $x^3-21x+20=(x-1)(x-4)(x+5)$

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How do you use the factor theorem to determine whether x-4 is a factor of $x^{3} - 21 x + 20$ $x^3-21x+20$ ?

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How do you use the factor theorem to determine whether x-4 is a factor of $x^{3} - 21 x + 20$ $x^3-21x+20$ ?