Question

Yes or no answer, determine weather binomial is a factor of polynomial

Answer 1

The Solution:

Given the polynomials and a binomial in each case as below:

$f(x)=2x^3+5x^2-37x-60;\text{ (x-4)}$$g(x)=3x^3-28x^2+29x+140;(x+7)$$h(x)=6x^5-15x^4-9x^3;(x+3)$

We are asked to determine whether each binomial is a factor in its case by answering Yes or No.

Note:

A binomial is an algebraic expression with two terms. If when equated to zero and the value put in the function yields zero, then the binomial is a factor, otherwise, it is not a factor.

So,

$f(x)=2x^3+5x^2-37x-60$

The binomial is:

$\begin{gathered} x-4=0 \\ x=4 \end{gathered}$

So,

$f(4)=2(4)^3+5(4)^2-37(4)-60=0$

Thus, Yes, x-4 is a factor of f(x).

$g(x)=3x^3-28x^2+29x+140$

The binomial is:

$\begin{gathered} x+7=0 \\ x=-7 \end{gathered}$

Substituting -7 for x in g(x).

$g(x)=3(-7)^3-28(-7)^2+29(-7)+140\ne0$

So, No, x+7 is Not a factor of g(x) because g(-7) is not equal to zero.

$h(x)=6x^5-15x^4-9x^3$

The binomial is:

$\begin{gathered} x+3 \\ x+3=0 \\ x=-1 \end{gathered}$

Substituting -3 for x in the function h(x), we get

$h(x)=6(-3)^5-15(-3)^4-9(-3)^3\ne0$

Thus, No, x+3 is Not a factor of g(x) because h(-3) is not equal to zero.