QuestionQuestion
Factor completely. $3 x^{2}-9 x-12$

Studdy Solution

STEP 1

1. The expression $3x^2 - 9x - 12$ is a quadratic polynomial.
2. We need to factor the expression completely, which involves finding the greatest common factor and factoring the quadratic expression.

STEP 2

1. Identify and factor out the greatest common factor (GCF) from the expression.
2. Factor the resulting quadratic expression.

STEP 3

Identify the greatest common factor (GCF) of the terms in the expression $3x^2 - 9x - 12$ .
The GCF of the coefficients $3, -9, \text{ and } -12$ is $3$ .
Factor out the GCF:
$3(x^2 - 3x - 4)$

STEP 4

Factor the quadratic expression $x^2 - 3x - 4$ .
To factor $x^2 - 3x - 4$ , look for two numbers that multiply to $-4$ (the constant term) and add to $-3$ (the coefficient of the linear term).
The numbers $-4$ and $1$ satisfy these conditions because:
$-4 \times 1 = -4$ $-4 + 1 = -3$
Rewrite the quadratic expression using these numbers:
$x^2 - 3x - 4 = (x - 4)(x + 1)$

STEP 5

Combine the factored terms with the GCF:
$3(x^2 - 3x - 4) = 3(x - 4)(x + 1)$
This is the completely factored form of the original expression.
The completely factored form of the expression is:
$\boxed{3(x - 4)(x + 1)}$

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