The special products can be used to perform selected multiplications. On the left, we use (x+y)(x-y) = x2-y2. On the right, (x-y)2 = x2-2xy+y2. Use special products to evaluate each expression. 1022

Ch. R - Review of Basic Concepts

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

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Lial 13th Edition

Ch. R - Review of Basic Concepts

Problem 115

Chapter 1, Problem 115

The special products can be used to perform selected multiplications.
On the left, we use (x+y)(x-y) = x²-y². On the right, (x-y)² = x²-2xy+y².

Use special products to evaluate each expression. 102²

Verified step by step guidance

Recognize that the expression 102^2 is a perfect square, which can be evaluated using the special product formula for the square of a binomial: $ (x + y)^2 = x^2 + 2xy + y^2 $.

Rewrite 102 as a sum of two numbers that are easier to square, for example, $ 102 = 100 + 2 $. Here, $ x = 100 $ and $ y = 2 $.

Apply the formula $ (x + y)^2 = x^2 + 2xy + y^2 $ by substituting $ x = 100 $ and $ y = 2 $:

\[ (100 + 2)^2 = 100^2 + 2 \times 100 \times 2 + 2^2 \]

Calculate each term separately: $ 100^2 $, $ 2 \times 100 \times 2 $, and $ 2^2 $, then add them together to find the value of $ 102^2 $.

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Key Concepts

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

Special Products Formulas

Special products are algebraic identities that simplify multiplication, such as the difference of squares (x + y)(x - y) = x² - y² and the square of a binomial (x - y)² = x² - 2xy + y². Recognizing these patterns helps quickly expand or simplify expressions without full multiplication.

Square of a Number Using Binomial Expansion

Squaring a number like 102² can be viewed as (100 + 2)², which fits the binomial square formula (a + b)² = a² + 2ab + b². This approach breaks down complex squares into simpler parts, making mental or written calculation easier and more efficient.

Applying Algebraic Identities to Numerical Expressions

Using algebraic identities to evaluate numerical expressions involves substituting numbers into formulas like special products. This method transforms arithmetic problems into algebraic ones, allowing for faster and more systematic calculations, especially with large or complex numbers.

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