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Chapter 0: Problem 67

Factor completely, or state that the polynomial is prime. $$4 x^{2}-4 x-24$$

Short Answer

Expert verified
The complete factorization of the given equation is \(4(x-3)(x+2)\).

Step by step solution

01

Identify the standard form of the Polynomial

The polynomial provided is \(4 x^{2}-4 x-24\). This is already in the standard form of a quadratic equation (\(ax^2+bx+c\)).
02

Factor out the Greatest Common Factor(GCF)

In looking at the terms, the greatest common factor is 4. Factoring out 4, we have \(4(x^2-x-6)\).
03

Apply Factoring Technique on Quadratic Inside Parentheses

The quadratic equation inside the parentheses can be factored as \((x-3)(x+2)\), since -3 and 2 are the roots of the equation \(x^2-x-6=0\), as they make the equation zero.
04

Write Complete Factorization

So the complete factorization of the given equation is \(4(x-3)(x+2)\).

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