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. 712

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 116

Chapter 1, Problem 116

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. 71²

Verified step by step guidance

Recognize that the expression 71^2 is a perfect square, which can be evaluated using the special product formula for a square of a binomial: \( (x - y)^2 = x^2 - 2xy + y^2 \).

Rewrite 71 as a number close to a convenient base, for example, 70 + 1, so that you can express 71^2 as \( (70 + 1)^2 \).

Apply the formula for the square of a sum: \( (x + y)^2 = x^2 + 2xy + y^2 \). Here, \(x = 70\) and \(y = 1\), so substitute these values into the formula.

Calculate each term separately: \( x^2 = 70^2 \), \( 2xy = 2 \times 70 \times 1 \), and \( y^2 = 1^2 \).

Combine all the terms to express 71^2 as \( 70^2 + 2 \times 70 \times 1 + 1^2 \), which simplifies the calculation using special products.

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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 evaluate expressions without full multiplication.

Difference of Squares

The difference of squares formula states that the product of a sum and difference of the same two terms equals the difference of their squares: (x + y)(x - y) = x² - y². This is useful for simplifying expressions and evaluating products efficiently.

Square of a Binomial

Squaring a binomial like (x - y)² expands to x² - 2xy + y². This formula helps in quickly finding the square of sums or differences without multiplying the binomial by itself step-by-step, saving time and reducing errors.

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