Question

Which of the following best describes the relationship between (x-3) and the polynomial x^3 + 4x^2 + 2?

A. (x-3) is not a factor
B. (x-3) is a factor
C. It is impossible to tell whether (x-3) is a factor

Answer 1

A. (x-3) is not a factor

Explanation:

You can find if (x-3) is a factor of the polynomial by dividing the polynomial by (x-3) by using long division or synthetic division.

Long division:

x^2+x+3

(x-3)/x^3+4x^2+0x+2

-(x^3-3x^2)

x^2+0x

-(x^2-3x)

3x+2

-(3x-9)

-7

Here you can see that (x-3) is not a factor of the polynomial because when you divide x^3 + 4x^2 + 2 by (x-3), there is a remainder of -7

Synthetic Division (A shortcut version of long division just to see if there is a remainder and if the supposed factor is really a factor) :

3 1 4 0 2

- 3 21 63

1 7 21 65

As seen before (x-3) is not a factor of the polynomial because there is a remainder. If 65 were 0, the (x-3) would be a factor of the polynomial.

Answer 2

A) (x-3) is not a factor of x^3+4x^2+2

Explanation:

(x-3) is a factor of f(x)=x^3+4x^2+2 if f(3)=0. This is by factor theorem.

So let's check it.

f(x)=x^3+4x^2+2

f(3)=3^3+4(3)^2+2

f(3)=27+4(9)+2

f(3)=27+36+2

f(3)=63+2

f(3)=65

Since f(3) doesn't equal 0, then x-3 is not a factor.