If $p(x)=x^3+x^2-4x-4$, then $p(x)$ is divisible by which of the following?

• $x+1$
• $x+2$
• $x+3$

Summary Submit

We can factor the polynomial by grouping.

$$p(x)=x^3+x^2-4x-4$$ $$p(x)=x^2(x+1)-4(x+1)$$ $$p(x)=(x^2-4)(x+1)$$ $$p(x)=(x-2)(x+2)(x+1)$$

In this form, we clearly see that it is divisible by $x+1$ and $x+2$.

If a binomial term is a factor of a polynomial, it refers to the zero or $x$-intercept of the polynomial. For example, if $x-a$ is a factor of a polynomial, $a$ must be a zero or $x$-intercept to the polynomial. We can check whether the three choices are $x$-intercepts:

$$p(x)=x^3+x^2-4x-4$$ $$0=x^3+x^2-4x-4$$

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$$x=-1\$$ $$0=(-1)^3+(-1)^2-4(-1)-4$$ $$0=-1+1+4-4$$ $$0=0   \checkmark$$

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$$x=-2\$$ $$0=(-2)^3+(-2)^2-4(-2)-4$$ $$0=-8+4+8-4$$ $$0=0   \checkmark$$

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$$x=-3\$$ $$0=(-3)^3+(-3)^2-4(-3)-4$$ $$0=-27+9+12-4$$ $$0=-10   ✖$$