Question

Is (x+5) a factor of f(x)=x^3 - 4x^2 + 3x + 7? Explain your reasoning

Answer 1

Explanation:

Consider the following polynomial:

$f\mleft(x\mright)=x^3-4x^2+3x+7$

suppose by contradiction that (x+5) is a factor of the given polynomial f(x). This means that:

$f\mleft(x\mright)=x^3-4x^2+3x+7=(x+5)Q(x)$

where Q(x) is another polynomial. Now, according to the above expression if we set f(x)=0, then we obtain:

$x^3-4x^2+3x+7=(x+5)Q(x)=0$

this is true when

$x+5=0$

that is, when:

$x=\text{ -5}$

this means that x= -5 is a root of f(x). In other words, this is the same to say that

$f\mleft(\text{ -5}\mright)=0$

But this is a contradiction since:

$f\mleft(\text{ -5}\mright)=(\text{ -5})^3-4(\text{ -5})^2+3(\text{ -5})+7=\text{ -233}\ne0$

then, we can conclude that the expression (x+5) is not a factor of f(x).

Thus, the correct answer is:

Answer:

Since f( -5) is not equal to 0, we can conclude that (x+5) is not a factor of f(x).