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Ch. R - Review of Basic Concepts
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. R - Review of Basic ConceptsProblem 107
Chapter 1, Problem 107

Factor by any method. See Examples 1–7. 64+(3x+2)3

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1
Recognize that the expression is a sum of two terms: \(64\) and \((3x+2)^3\). Notice that \(64\) is a perfect cube since \(64 = 4^3\).
Rewrite the expression as a sum of cubes: \(4^3 + (3x+2)^3\).
Recall the sum of cubes factoring formula: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$.
Identify \(a = 4\) and \(b = (3x+2)\), then apply the formula: \((4 + (3x+2)) \left(4^2 - 4(3x+2) + (3x+2)^2\right)\).
Simplify the first factor and expand the terms inside the second factor to write the fully factored form.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Sum of Cubes Formula

The sum of cubes formula states that a³ + b³ = (a + b)(a² - ab + b²). It is used to factor expressions where two terms are each perfect cubes added together. Recognizing the expression as a sum of cubes allows for straightforward factoring.
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Special Products - Cube Formulas

Identifying Perfect Cubes

To apply the sum of cubes formula, each term must be a perfect cube. For example, 64 is 4³, and (3x + 2)³ is already a cube. Recognizing these cubes is essential to rewrite the expression in a form suitable for factoring.
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Polynomial Factoring Techniques

Factoring polynomials involves rewriting expressions as products of simpler polynomials. Techniques include factoring by grouping, special formulas like sum/difference of cubes, and recognizing patterns. Choosing the correct method simplifies solving or further manipulation.
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Introduction to Factoring Polynomials
Related Practice
Textbook Question

Perform the indicated operations. Assume all variables represent positive real numbers. 3x5y4−2xxy43\(\sqrt\)[4]{x^5y} - 2x\(\sqrt\)[4]{xy}

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Textbook Question

Let U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, M = {0, 2, 4, 6, 8}, N = {1, 3, 5, 7, 9, 11, 13}, Q = {0, 2, 4, 6, 8, 10, 12}, and R = {0, 1, 2, 3, 4}.Use these sets to find each of the following. Identify any disjoint sets. {x | x ∈ M and x ∈ Q}

Textbook Question

Multiply or divide as indicated. 55.76 * 72

Textbook Question

Evaluate each expression for p=−4p=-4, q=8q=8, and r=−10r=-10. q2−r33p4+q8\(\frac{\frac{q}{2}\)-\(\frac{r}{3}\)}{\(\frac{3p}{4}\)+\(\frac{q}{8}\)}

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Textbook Question

Multiply or divide as indicated. 22.41 × 33

Textbook Question

Let U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, M = {0, 2, 4, 6, 8}, N = {1, 3, 5, 7, 9, 11, 13}, Q = {0, 2, 4, 6, 8, 10, 12}, and R = {0, 1, 2, 3, 4}.Use these sets to find each of the following. Identify any disjoint sets. {x | x ∈ M or x ∈ Q}