State if the give binomial is a factor of the given polynomial (9x ^(3) + 57x^(2) + 21x + 24) / (x + 6)

Question

State if the give binomial is a factor of the given polynomial
`(9x ^(3) + 57x^(2) + 21x + 24) / (x + 6)`

Answer 1

We want to find out if (x+6) is a factor of the polynomial

`9x^3+57x^2+21x+24`

In order to find this, we can use the factor theorem.

If we have a polynomial f(x) and want to find if (x-a) is a factor of this polynomial, we plug in x = a into the function and if we get 0, (x-a) is a factor(!)

Now, let's plug in:

x = -6 into the polynomial and see if we get a 0 or not.

Steps shown below:

`\begin{gathered} 9x^3+57x^2+21x+24 \\ 9(-6)^3+57(-6)^2+21(-6)+24 \\ =-1944+2052-126+24 \\ =6 \end{gathered}`

Answer

Since it doesn't produce a 0, (x + 6 ) is not a factor of the polynomial given.