Problem 50 Factor completely. $$ x^{3}+... [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 10: Problem 50

Factor completely. $$ x^{3}+x^{2}+3 x+3 $$

Short Answer

Expert verified

$(x + 1)(x^2 + 3)$

Step by step solution

Group the terms

Group the terms in pairs that can be factored separately: \[x^3 + x^2 + 3x + 3 = (x^3 + x^2) + (3x + 3)\]

Factor out the greatest common factor (GCF) in each pair

Factor out the GCF from each of the grouped pairs: \[x^2(x + 1) + 3(x + 1)\]

Factor by grouping

Notice that each term now has a common binomial factor $(x + 1)$. Factor this out: \[(x^2 + 3)(x + 1)\]

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Key Concepts

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

Grouping Method

To factor a polynomial like $ x^{3} + x^{2} + 3x + 3 $, the grouping method can be very useful. This method involves rearranging and grouping the terms to make the polynomial easier to factor.
The first step is to separate the polynomial into smaller groups. For the example polynomial, it is split like this:
\[ x^{3} + x^{2} + 3x + 3 = (x^{3} + x^{2}) + (3x + 3) \]
By grouping the terms into pairs, we can factor each pair individually. This approach simplifies the polynomial, paving the way for the application of other factoring techniques in subsequent steps.

Greatest Common Factor (GCF)

After using the grouping method, the next vital step is to identify and factor out the Greatest Common Factor (GCF) from each group. The GCF is the largest factor that divides all terms in a group without leaving a remainder. For example:
In the first group $x^{3} + x^{2}$, the GCF is $x^{2}$: \[ x^{2}(x + 1) \]
In the second group $3x + 3$, the GCF is $3 $: \[ 3(x + 1) \]
Applying these factors we get: \[ x^{2}(x + 1) + 3(x + 1) \]
Factoring out the GCF helps to simplify and organize the polynomial, making it more manageable to solve.

Binomial Factorization

The polynomial is now simplified to a form where each term contains a common binomial factor. In $x^{2}(x + 1) + 3(x + 1)$, the binomial factor $x + 1$ appears in both terms. We can factor out this binomial to further simplify: \[ (x + 1)(x^{2} + 3) \]
This step, known as binomial factorization, involves identifying and extracting common binomial factors from grouped terms. Through binomial factorization, the original polynomial has been completely factored to $ (x + 1)(x^{2} + 3) $.
This technique highlights the importance of recognizing common patterns and factors within polynomials, significantly simplifying seemingly complex algebraic expressions.

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91影视

Factor completely. $$ x^{3}+x^{2}+3 x+3 $$

Short Answer

Step by step solution

Group the terms

Factor out the greatest common factor (GCF) in each pair

Factor by grouping

Key Concepts

Grouping Method

Greatest Common Factor (GCF)

Binomial Factorization

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