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Chapter 10: Problem 55

Multiply and simplify. Assume all variables represent nonnegative real numbers. $$(c+9)(c-9)$$

Short Answer

Expert verified
The short answer is: \(c^2 - 81\)

Step by step solution

01

Identify the binomials

We have two binomials to multiply: \((c+9)\) and \((c-9)\).
02

Use the distributive property (FOIL method)

The FOIL method stands for First, Outer, Inner, Last. We'll multiply the terms accordingly: First terms: \(c * c = c^2\) Outer terms: \(c * -9 = -9c\) Inner terms: \(9 * c = 9c\) Last terms: \(9 * -9 = -81\)
03

Combine the terms

Now, we need to combine the terms obtained in the previous step: \(c^2 - 9c + 9c - 81\)
04

Simplify the expression

We can simplify the expression by adding the like terms: \(c^2 - 9c + 9c - 81 = c^2 - 81\)
05

Write the final simplified expression

The final simplified expression is: \(c^2 - 81\)

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Key Concepts

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

Understanding Binomials
A binomial is an algebraic expression that consists of two distinct terms separated by either addition or subtraction. In the exercise given, we are working with two binomials:
  • (c + 9)
  • (c - 9)
The components of a binomial are important because they provide a clear structure for how the terms are combined. This understanding is essential when you're about to multiply the binomials using certain algebraic techniques, such as the FOIL method. Each term in the binomial plays a specific role in the calculations that follow.
The Essence of the Distributive Property
The distributive property is a fundamental concept in algebra used to simplify expressions, especially when dealing with multiplication across parentheses. It states that multiplying a sum by a number is the same as multiplying each addend individually and then adding the products. This principle can be expressed as:
  • \( a(b + c) = ab + ac \)
When you encounter algebraic expressions like binomials, the distributive property helps break down these expressions into more manageable pieces. Using the distributive property is crucial for performing operations like the FOIL method, which involves distributing each term across the other.
Understanding and Using the FOIL Method
The FOIL method is a handy mnemonic used to remember the order of operations when multiplying two binomials. The term FOIL stands for:
  • First
  • Outer
  • Inner
  • Last
Each word corresponds to the term pairs you multiply together:1. **First:** Multiply the first terms in each binomial; - \( c * c = c^2 \)2. **Outer:** Multiply the terms on the outer ends of the expression; - \( c * -9 = -9c \)3. **Inner:** Multiply the inner terms; - \( 9 * c = 9c \)4. **Last:** Multiply the last terms in each binomial. - \( 9 * -9 = -81 \)Using the FOIL method ensures that every part of each binomial is accurately multiplied, providing you with all the components needed before simplification.
The Process of Simplification
Simplification in algebra is the process of combining like terms to create a more concise and understandable expression. After using the FOIL method, you get terms that need to be simplified:- \( c^2 - 9c + 9c - 81 \)Notice here, the terms \(-9c\) and \(9c\) are like terms. They cancel each other out when added together. This step reduces the expression to:- \( c^2 - 81 \)This simplified expression is easier to handle and interpret, giving you the final result in its most straightforward form. Simplifying expressions allows for cleaner and more intuitive conclusions in algebra.

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Most popular questions from this chapter

Simplify completely. $$\frac{30-18 \sqrt{5}}{4}$$

Rationalize the denominator of each expression. Assume all variables represent positive real numbers. $$\sqrt[4]{\frac{2}{3 t^{2}}}$$

Multiply and simplify. $$\sqrt{\frac{10}{7}} \cdot \sqrt{\frac{7}{3}}$$

How do you find the conjugate of a binomial?

Rationalize the denominator and simplify completely. Assume the variables represent positive real numbers. $$\frac{\sqrt{32}}{\sqrt{5}-\sqrt{7}}$$
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