Answer
$y = x^2(x + 4)(x - 1)$
Work Step by Step
Factor out the greatest common factor, which is $x^2$, from all terms:
$y = x^2(x^2 + 3x - 4)$
The expression within the parentheses is now in standard form, which is given by the formula:
$ax^2 + bx + c$, where $a$ is the coefficient of the squared term, $b$ is the coefficient of the linear term, and $c$ is the constant term.
Factor the expression. Find factors that when multiplied together will equal $ac$ but when added together will equal $b$. The $a$ is the coefficient of the first term, $b$ is the coefficient of the second term, and $c$ is the constant term.
In this expression, $a$ is $1$, $b$ is $3$, and $c$ is $-4$; therefore, $ac$ is $-4$. By looking at $ac$ and $b$, we see that the factor with the greater absolute value must be positive and the one with the smaller absolute value should be negative.
The possibilities are:
$4$ and $-1$
$2$ and $-2$
The first option will work.
Rewrite the equation in factor form:
$y = x^2(x + 4)(x - 1)$
Check this answer using the FOIL method to expand the binomial.
Expand the expression by multiplying the terms within them according to the FOIL method, meaning the first terms are multiplied first, then the outer, then the inner, and, finally, the last terms:
$y = x^2[(x)(x) + (x)(-1) + (4)(x) + (4)(-1)]$
Multiply to simplify:
$y = x^2(x^2 - x + 4x - 4)$
Combine like terms:
$y = x^2(x^2 + 3x - 4)$
Use distributive property:
$y = (x^2)(x^2) + (x^2)(3x) + (x^2)(-4)$
Multiply to simplify:
$y = x^4 + 3x^3 - 4x^2$
This is the same function that we started out with. Therefore, the factoring was correct.
