How do you use the factor theorem to determine whether x+1 is a factor of $x^{3} + x^{2} + x + 1$ $x^3 + x^2 + x + 1$ ?

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How do you use the factor theorem to determine whether x+1 is a factor of $x^{3} + x^{2} + x + 1$ $x^3 + x^2 + x + 1$ ?

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Answer 1 · 2016-01-03T06:38:35

The factor theorem states that a polynomial $f (x)$ $f(x)$ has a factor $(x + k)$ $(x+k)$ if and only if $f (- k) = 0$ $f(-k)=0$ .
Here $x^{3} + x^{2} + x + 1$ $x^3+x^2+x+1$ is a polynomial.
Let $f (x) = x^{3} + x^{2} + x + 1$ $f(x)=x^3+x^2+x+1$

Now we want to know that is $x + 1$ $x+1$ a factor of $f (x)$ $f(x)$ or not.

For this purpose we have to put $x = - 1$ $x=-1$ in $f (x)$ $f(x)$ , if the result comes to be $0$ $0$ then $x + 1$ $x+1$ is a factor of $f (x)$ $f(x)$ and if the result comes not to be $0$ $0$ then $x + 1$ $x+1$ is not a factor of $f (x)$ $f(x)$ .

Put $x = - 1$ $x=-1$ in $f (x)$ $f(x)$
$\Rightarrow f (- 1) = {(- 1)}^{3} + {(- 1)}^{2} + (- 1) + 1 = - 1 + 1 - 1 + 1 = 0$ $implies f(-1)=(-1)^3+(-1)^2+(-1)+1=-1+1-1+1=0$

Since the result is $0$ $0$ , therefore $x + 1$ $x+1$ is a factor of the given polynomial.

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How do you use the factor theorem to determine whether x+1 is a factor of $x^{3} + x^{2} + x + 1$ $x^3 + x^2 + x + 1$ ?

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