Question

Given 5x3+8x2−7x−6. 1) The binomial (x+2) is a factor of the polynomial expression. Describe how you know it is a factor. 2) The binomial (x+1) is NOT a factor of the polynomial expression. Explain how you know it is not a factor.

Answer 1

ANSWER TO QUESTION 1

Given

`f(x)=5x^3+8x^2-7x-6`

We can use the factor theorem to determine if

`x+2` is a factor of the polynomial or not.

According to this theorem, if
`x+2` is a factor of
`f(x)`, then
`f(-2)=0`.

How did we get the
`-2`?

We set
`x+2=0` and then solve to obtain
`x=-2`.

So now let us plug in
`x=-2` in to the function to see if it will simplify to zero.

`f(-2)=5(-2)^3+8(-2)^2-7(-2)-6`

`f(-2)=5(-8)+8(4)+7(2)-6`

`f(-2)=-40+32+14-6`

`f(-2)=6-6`

`f(-2)=0`

Since the result simplifies to zero, we conclude that

`x+2` is a factor of

`f(x)=5x^3+8x^2-7x-6`

ANSWER TO QUESTION 2

We have the function,

`f(x)=5x^3+8x^2-7x-6`

We can use the remainder theorem to show that

`x+1` is NOT a factor of the polynomial.

According to this theorem, if
`x+1` is not a factor of
`f(x)`, then
`f(-1)\\e 0`.

So now let us plug in
`x=-1` in to the function to see if it will simplify to non-zero number.

`f(-1)=5(-1)^3+8(-1)^2-7(-1)-6`

`f(-1)=5(-1)+8(1)+7(1)-6`

`f(-1)=-5+8+7-6`

`f(-1)=4`

`f(-1)\\e0`

Since the result simplifies to a non zero number, we conclude that

`x+1` is NOT a factor of

`f(x)=5x^3+8x^2-7x-6`