Factor each polynomial. See Example 7. (5x-2)3-8

Ch. R - Review of Basic Concepts

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

All textbooks

Lial 13th Edition

Ch. R - Review of Basic Concepts

Problem 90

Chapter 1, Problem 90

Factor each polynomial. See Example 7. (5x-2)³-8

Verified step by step guidance

Recognize that the expression $ (5x - 2)^3 - 8 $ is a difference of cubes, since $8$ can be written as \$2^3$.

Recall the difference of cubes formula: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$.

Identify $a = (5x - 2)$ and $b = 2$ in the expression.

Apply the formula: write the factorization as $((5x - 2) - 2)((5x - 2)^2 + (5x - 2)(2) + 2^2)$.

Simplify each factor: first simplify $((5x - 2) - 2)$, then expand and simplify the quadratic expression inside the second factor.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

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

Difference of Cubes

The difference of cubes formula states that a³ - b³ = (a - b)(a² + ab + b²). It is used to factor expressions where two perfect cubes are subtracted. Recognizing the structure helps simplify polynomials like (5x - 2)³ - 8 by identifying a = (5x - 2) and b = 2.

Polynomial Factoring

Polynomial factoring involves rewriting a polynomial as a product of simpler polynomials. This process simplifies expressions and solves equations. Factoring techniques include recognizing special products like difference of squares, sum/difference of cubes, and factoring by grouping.

Exponents and Powers

Understanding exponents is crucial for identifying perfect cubes and manipulating expressions like (5x - 2)³. Exponents indicate repeated multiplication, and recognizing powers helps in applying formulas such as the difference of cubes for factoring.

Recommended video:

04:10

Powers of i

Related Practice

Textbook Question

Evaluate each expression. 9∙3 - 16÷4

Textbook Question

Factor each polynomial. See Example 7. (3x+4)³-1

Textbook Question

Simplify each radical. Assume all variables represent positive real numbers. ⁸√5⁴

views

Textbook Question

Write each decimal as a fraction. (Do not write in lowest terms.) 0.043

Textbook Question

Write each decimal as a fraction. (Do not write in lowest terms.) 0.087

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. Q′

views