Skip to main content
Ch. P - Fundamental Concepts of Algebra
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. P - Fundamental Concepts of AlgebraProblem 21
Chapter 1, Problem 21

Multiply or divide as indicated. x3−8x2−4⋅x+23x\(\frac{x^3 - 8}{x^2 - 4}\) \(\cdot\) \(\frac{x + 2}{3x}\)

Verified step by step guidance
1
Identify the given expression to multiply: \(\frac{\left(x^3 - 8\right)}{\left(x^2 - 4\right)} \cdot \frac{\left(x + 2\right)}{3x}\).
Factor all polynomials where possible. Recognize that \(x^3 - 8\) is a difference of cubes and \(x^2 - 4\) is a difference of squares. Use the formulas: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\) and \(a^2 - b^2 = (a - b)(a + b)\).
Rewrite the expression with factored forms: \(\frac{(x - 2)(x^2 + 2x + 4)}{(x - 2)(x + 2)} \cdot \frac{(x + 2)}{3x}\).
Cancel out common factors in the numerator and denominator, such as \((x - 2)\) and \((x + 2)\), to simplify the expression.
Multiply the remaining factors in the numerator and denominator to write the simplified expression as a single fraction.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2m
Was this helpful?

Key Concepts

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

Polynomial Factoring

Factoring polynomials involves rewriting expressions as products of simpler polynomials. Recognizing special forms like difference of cubes (x³ - 8) and difference of squares (x² - 4) helps simplify expressions before multiplication or division.
Recommended video:
Guided course
07:30
Introduction to Factoring Polynomials

Multiplication and Division of Rational Expressions

When multiplying or dividing rational expressions, factor all numerators and denominators first, then multiply across numerators and denominators. For division, multiply by the reciprocal of the divisor to simplify the expression.
Recommended video:
Guided course
02:58
Rationalizing Denominators

Simplifying Rational Expressions

Simplifying involves canceling common factors in the numerator and denominator after factoring. This reduces the expression to its simplest form, making it easier to interpret or use in further calculations.
Recommended video:
Guided course
05:07
Simplifying Algebraic Expressions
Related Practice
Textbook Question

Simplify the algebraic expression. (1/5)(5x) + [(3y) + (- 3y)] - (-x)

1100
views
Textbook Question

Evaluate each exponential expression in Exercises 1–22. 23/272^3/2^7

753
views
Textbook Question
Simplify the algebraic expression. 5(2x - 3) + 7x
1096
views
1
rank
Textbook Question

Use the product rule to simplify the expressions in Exercises 13–22. In Exercises 17–22, assume that variables represent nonnegative real numbers. 6x⋅3x2\(\sqrt{6x}\[\cdot\]\sqrt{3x^2}\)

1266
views
Textbook Question

Find the intersection of the sets. {1,2,3,4}∩{2,4,5}

1056
views
Textbook Question

Use the product rule to simplify the expressions in Exercises 13–22. In Exercises 17–22, assume that variables represent nonnegative real numbers. 2x2⋅6x\(\sqrt{2x^2}\[\cdot\]\sqrt{6x}\)

587
views