Ch. R - Review of Basic Concepts

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

Lial 13th Edition

Ch. R - Review of Basic Concepts

Problem 53

Chapter 1, Problem 53

Find each product. [(3q+5)-p][(3q+5)+p]

Verified step by step guidance

Recognize that the expression $[(3q+5)-p][(3q+5)+p]$ is in the form of a product of conjugates, which follows the pattern $(a - b)(a + b) = a^2 - b^2$.

Identify $a = (3q + 5)$ and $b = p$ from the given expression.

Apply the difference of squares formula: $(3q + 5)^2 - p^2$.

Expand $(3q + 5)^2$ by using the formula $(x + y)^2 = x^2 + 2xy + y^2$, where $x = 3q$ and $y = 5$.

Write the final expression as $ (3q)^2 + 2 \cdot 3q \cdot 5 + 5^2 - p^2 $ and simplify each term without calculating the final numeric values.

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 Squares

The difference of squares is a special product formula expressed as (a - b)(a + b) = a² - b². It allows quick multiplication of two binomials that are conjugates, simplifying the expression by subtracting the square of the second term from the square of the first.

Binomial Expressions

A binomial is an algebraic expression containing two terms connected by addition or subtraction, such as (3q + 5) or p. Understanding how to manipulate and multiply binomials is essential for expanding and simplifying algebraic expressions.

Polynomial Multiplication

Polynomial multiplication involves applying the distributive property to multiply each term in one polynomial by every term in the other. This process is fundamental for expanding products of binomials and simplifying the resulting expressions.

Recommended video:

03:42

Finding Zeros & Their Multiplicity

Related Practice

Textbook Question

Simplify each expression. Write answers without negative exponents. Assume all variables represent nonzero real numbers. y⁸/y¹²

Textbook Question

Find each product or quotient where possible. 5(-7)

views

Textbook Question

Simplify each expression. Write answers without negative exponents. Assume all variables represent nonzero real numbers. r⁷/r¹⁰

Textbook Question

Add or subtract, as indicated. 1/6m + 2/5m + 4/m

views

Textbook Question

Let A = {2, 4, 6, 8, 10, 12}, B = {2, 4, 8, 10}, C = {4, 10, 12}, D = {2, 10}, andU = {2, 4, 6, 8, 10, 12, 14}. Determine whether each statement is true or false. D ⊆ A

Textbook Question

Factor each trinomial, if possible. See Examples 3 and 4. $32 a^{2} + 48 a b + 18 b^{2}$