Question

select the correct answer.
what is the completely factored form of the polynomial?
$5x^4 + 20x^3 - 105x^2$
a. $5x^2(x + 3)(x - 7)$
b. $5x^3(x + 3)(x - 7)$
c. $5x^2(x - 3)(x + 7)$
d. $5x^3(x - 3)(x + 7)$

Explanation:

Step1: Factor out the GCF

The greatest common factor (GCF) of \(5x^4\), \(20x^3\), and \(-105x^2\) is \(5x^2\). Factoring out \(5x^2\) gives:
\(5x^2(x^2 + 4x - 21)\)

Step2: Factor the quadratic

We need to factor \(x^2 + 4x - 21\). We look for two numbers that multiply to \(-21\) and add to \(4\). The numbers are \(7\) and \(-3\) (since \(7 \times (-3) = -21\) and \(7 + (-3) = 4\)). So,
\(x^2 + 4x - 21 = (x + 7)(x - 3)\)

Step3: Combine the factors

Putting it all together, the completely factored form is:
\(5x^2(x - 3)(x + 7)\)

Answer:

C. \(5x^2(x - 3)(x + 7)\)