Problem 49 Write an equivalent expression b... [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 49

Write an equivalent expression by factoring. $$t^{3}+6 t^{2}-2 t-12$$

Short Answer

Expert verified

The equivalent expression by factoring is $(t + 6)(t^{2} - 2)$.

Step by step solution

Group the terms

Group the given polynomial in pairs to simplify the factoring process. Group the polynomial as \[ t^{3}+6 t^{2}-2 t-12 = (t^{3} + 6t^{2}) + (-2t - 12) \]

Factor out the greatest common factor (GCF) from each group

Identify and factor out the GCF from each pair. For $t^{3} + 6t^{2}$, the GCF is $ t^{2} $. For $-2t - 12$, the GCF is $-2$. So it becomes: \[t^{2}(t + 6) - 2(t + 6)\]

Factor by grouping

Since both terms contain a common binomial factor $(t + 6)$, factor this binomial out. \[t^{2}(t + 6) - 2(t + 6) = (t + 6)(t^{2} - 2)\]

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Key Concepts

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

grouping method

The grouping method is a helpful strategy for factoring polynomials, especially when dealing with four terms. This method involves rearranging and grouping the terms into pairs. Once grouped, each pair is factored separately.

Let's consider our example:
The polynomial given is: \[ t^{3}+6 t^{2}-2 t-12 \]
We start by grouping the terms: \[ t^{3}+6 t^{2} - 2 t - 12 \]
We can form two groups as follows: \[ (t^{3} + 6t^{2}) + (-2t - 12) \]
Grouping ensures we can later identify common factors within these newly created smaller polynomials.

greatest common factor (GCF)

The greatest common factor (GCF) is the largest number or expression that divides each term in a polynomial without leaving a remainder. Finding the GCF is crucial because it simplifies further calculations and helps in factoring.

In our example:
For the group $ (t^{3} + 6t^{2}) $, the GCF is $ t^{2} $
For the group $ (-2t - 12) $, the GCF is $ -2 $.
By factoring out the GCF from each group, we have: $ t^{2}(t + 6) $ and $ -2(t + 6) $.

factoring by grouping

Factoring by grouping takes advantage of the common factors found in each pair of grouped terms. After factoring out the GCF, you often find a common binomial that can be factored out again.

In the previous steps, we factored out $ t^{2} $ and $ -2 $, giving us: \[ t^{2}(t + 6) - 2(t + 6) \]
Notice that $ (t + 6) $ is a common factor. Hence, we can factor it out: \[ (t + 6)(t^{2} - 2) \]
This leaves us with the factored form of the original polynomial. Factoring by grouping simplifies complex polynomials by methodically reducing them step-by-step.

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91影视

Write an equivalent expression by factoring. $$t^{3}+6 t^{2}-2 t-12$$

Short Answer

Step by step solution

Group the terms

Factor out the greatest common factor (GCF) from each group

Factor by grouping

Key Concepts

grouping method

greatest common factor (GCF)

factoring by grouping

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