Question

Using the Factor theorem to determine if g(x) is a factor of f(x), where f(x)=x5+3x4−x3−3x2+5x+15 and g(x)=x+3.

Answer 1

To determine if g(x)=x+3 is a factor of f(x)=x^5+3x^4-x^3-3x^2+5x+15, we can substitute -3 for x in f(x) and check if the result is zero. If it is, then g(x) is a factor of f(x).

Step-by-step explanation:

To determine if g(x)=x+3 is a factor of f(x)=x^5+3x^4-x^3-3x^2+5x+15, we can use the Factor theorem. According to the Factor theorem, if g(x) is a factor of f(x), then when we substitute -3 for x in f(x), the result should be zero. Let's substitute -3 for x in f(x) and check if the result is zero:

f(-3) = (-3)^5+3(-3)^4-(-3)^3-3(-3)^2+5(-3)+15

Simplifying the expression, we get f(-3) = -243-243+27-27-15+15 = -486.

Since f(-3) is not zero, g(x)=x+3 is not a factor of f(x).