Question

lorelei claims that (x + 3) is a factor of p(x)=x^4 + 5x^3+x^2 - 11x + 12. which of the following methods could she use to prove her claim? select three that apply. a show that the value of p(3) is 0 b show that the quotient of \\(\frac{p(x)}{x + 3}\\) is a polynomial c show that p(x)=(x + 3)·q(x) for some polynomial q(x) d show that the remainder when p(x) is divided by (x + 3) is 0

Explanation:

Step1: Recall factor - theorem

According to the factor - theorem, if $(x - a)$ is a factor of a polynomial $p(x)$, then $p(a)=0$. Here, if $(x + 3)$ is a factor of $p(x)$, we set $x+3 = 0$, so $x=-3$ and we should check $p(-3)=0$, not $p(3)=0$. So option A is incorrect.

Step2: Recall polynomial division

If $(x + 3)$ is a factor of $p(x)$, then by the division algorithm for polynomials, $p(x)=(x + 3)\cdot q(x)$ for some polynomial $q(x)$. So option C is correct.

Step3: Recall the form of quotient

When we divide a polynomial $p(x)$ by $(x + 3)$, if $(x + 3)$ is a factor, the quotient $\frac{p(x)}{x + 3}=q(x)$ where $q(x)$ is a polynomial. So option B is correct.

Step4: Recall the remainder - theorem

By the remainder - theorem, when $p(x)$ is divided by $(x + 3)$ (i.e., $x-(-3)$), the remainder $r=p(-3)$. If $(x + 3)$ is a factor, then the remainder $r = 0$. So option D is correct.

Answer:

B. show that the quotient of $\frac{p(x)}{x + 3}$ is a polynomial
C. show that $p(x)=(x + 3)\cdot q(x)$ for some polynomial $q(x)$
D. show that the remainder when $p(x)$ is divided by $(x + 3)$ is 0