Problem 95 Factor completely. $$ 3(x+1)... [FREE SOLUTION]

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Intermediate Algebra : Concepts and Applications

Marvin L. Bittinger, David J. Ellenbogen, Barbara L. Johnson

$Math Studyset 91影视 Explanations$ Math

10 Edition

Chapter 5: Problem 95

Factor completely. $$ 3(x+1)^{2}-9(x+1)-12 $$

Short Answer

Expert verified

$3(x+2)(x-3)$

Step by step solution

Identify Common Terms

Recognize that each term in the expression contains a factor of $x+1$. Rewrite the expression using the common factor: \[3(x+1)^2 - 9(x+1) - 12\]

Introduce a Substitution

Let $y = x + 1$. Substitute $y$ into the expression to simplify it: \[3y^2 - 9y - 12\]

Factor the Quadratic Expression

Factor the quadratic expression $3y^2 - 9y - 12$. First, factor out the greatest common factor (GCF), if any. In this case, we can directly factorize it as: \[3(y+1)(y-4)\]

Substitute Back

Replace $y$ with $x+1$ to revert to the original variable: \[3((x+1)+1)((x+1)-4)\]

Simplify the Factors

Simplify the factors to obtain the final expression: \[3(x+2)(x-3)\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Factor

Factoring a polynomial often starts by identifying a common factor within the terms. In the given problem, we see that each term contains $(x+1)$. Identifying this common factor is a crucial step.
For example:

$3(x+1)^2$
$-9(x+1)$
$-12$

Each of these terms shares the common factor $(x+1)$. By recognizing this, we simplify the initial polynomial as:
\[3(x+1)^2 - 9(x+1) - 12\]. This simplification makes further steps easier.

Substitution Method

The substitution method is used to simplify complex expressions. By substituting a variable, we make it easier to handle.
In this exercise, let鈥檚 substitute $y = x + 1$. The expression then transforms:
\[3(x+1)^2 - 9(x+1) - 12 \rightarrow 3y^2 - 9y - 12\].
This new expression is simpler to work with. Substitution helps in avoiding errors and makes the factoring process efficient.

Quadratic Expression

A quadratic expression is in the form $ax^2 + bx + c$. In our problem, $3y^2 - 9y - 12$ is a quadratic expression.
To factorize this, we look for two numbers that multiply to $a*c = 3*(-12) = -36$ and add to $-9$. These numbers are $3$ and $-12$. Therefore, we rewrite and factor as:
\[3(y+1)(y-4)\].
Finally, substituting back, we get:
\[3((x+1)+1)((x+1)-4) \rightarrow 3(x+2)(x-3)\].
This final expression represents the completely factored form of the original polynomial.

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91影视

Factor completely. $$ 3(x+1)^{2}-9(x+1)-12 $$

Short Answer

Step by step solution

Identify Common Terms

Introduce a Substitution

Factor the Quadratic Expression

Substitute Back

Simplify the Factors

Key Concepts

Common Factor

Substitution Method

Quadratic Expression

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