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

Yes or no answer, determine weather binomial is a factor of polynomial-example-1

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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\\e0

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\\e0

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

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