Answer :

We want to factor the polynomial

[tex]$$
4x^4 - 28x^3 - 120x^2.
$$[/tex]

Step 1. Factor out the Greatest Common Factor (GCF):

Each term in the polynomial has a common factor of [tex]$4x^2$[/tex]. Factoring this out, we have

[tex]$$
4x^4 - 28x^3 - 120x^2 = 4x^2 \left( x^2 - 7x - 30 \right).
$$[/tex]

Step 2. Factor the Quadratic Expression:

We now focus on the quadratic expression

[tex]$$
x^2 - 7x - 30.
$$[/tex]

To factor this, we need to find two numbers whose product is [tex]$-30$[/tex] and whose sum is [tex]$-7$[/tex]. The two numbers that satisfy these conditions are [tex]$-10$[/tex] and [tex]$3$[/tex], since

[tex]$$
-10 \times 3 = -30 \quad \text{and} \quad -10 + 3 = -7.
$$[/tex]

Thus, the quadratic factors as

[tex]$$
x^2 - 7x - 30 = (x - 10)(x + 3).
$$[/tex]

Step 3. Write the Completely Factored Form:

Substitute the factorization of the quadratic back into the expression:

[tex]$$
4x^4 - 28x^3 - 120x^2 = 4x^2 (x - 10)(x + 3).
$$[/tex]

This is the completely factored form of the given polynomial.

Other Questions