Question

Is the binomial a factor of the polynomial function?f (x) = x^3 + 4x^2 - 9x - 36

Answer 1

For the binomials (x - 3), (x + 3), and (x + 4), you would select "Yes," and for (x - 1) and (x - 4), you would select "No."

To determine if each binomial is a factor of the polynomial function f(x) = x^3 + 4x^2 - 9x - 36 , use the Factor Theorem. According to the Factor Theorem, a binomial (x - a) is a factor of a polynomial if and only if f(a) = 0.

By evaluating f(x) at the value that would make the binomial zero. For example, for (x - 1), we would test f(1) , for (x - 3) , we would test f(3) , and so on.

Based on the calculations, the results for each binomial are:

• (x - 1) is not a factor of the polynomial f(x).
• (x - 3) is a factor of the polynomial f(x).
• (x + 3) is a factor of the polynomial f(x).
• (x - 4) is not a factor of the polynomial f(x).
• (x + 4) is a factor of the polynomial f(x).

Answer 2

In order to find if each binomial is a factor of the given polynomial function, we can use the zero remainder theorem.

This theorem states that if a value x = k is a zero of the function f(x) (that is, f(k) = 0), so the binomial (x - k) is a factor of the function f(x).

Therefore, let's check the values 1, 3, -3, 4 and -4:

`\begin{gathered} f(1)=1+4-9-36=-40 \\ f(3)=27+36-27-36=0 \\ f(-3)=-27+36+27-36=0 \\ f(4)=64+64-36-36=64 \\ f(-4)=-64+64+36-36=0 \end{gathered}`

Therefore the answer is:

(x - 1): No

(x - 3): Yes

(x + 3): Yes

(x - 4): No

(x + 4): Yes