Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

0. Review of Algebra

Multiplying Polynomials

College Algebra

0. Review of Algebra

Multiplying Polynomials

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2:11 minutes

Problem 114

Textbook Question

The special products can be used to perform selected multiplications. On the left, we use (x+y)(x-y) = x^2-y^2. On the right, (x-y)^2 = x^2-2xy+y^2. Use special products to evaluate each expression. x = 63 y = 57

Verified step by step guidance

Identify the expressions to evaluate: (x+y)(x-y) and (x-y)^2.

Substitute the given values: x = 63 and y = 57 into the expressions.

For (x+y)(x-y), use the identity: (x+y)(x-y) = x^2 - y^2.

Calculate x^2 and y^2 separately using the substituted values.

For (x-y)^2, use the identity: (x-y)^2 = x^2 - 2xy + y^2 and substitute the values to find each term.

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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 that states (a + b)(a - b) = a² - b². This identity allows for the simplification of expressions involving the multiplication of two binomials where one is the sum and the other is the difference of the same two terms. It is particularly useful for quickly calculating the product of two numbers that are equidistant from a central value.

Perfect Square Trinomials

A perfect square trinomial is formed when a binomial is squared, resulting in the expansion (a ± b)² = a² ± 2ab + b². This concept is essential for simplifying expressions where a binomial is multiplied by itself, allowing for the identification of the squared terms and the middle term that is twice the product of the two terms in the binomial.

Substitution in Algebraic Expressions

Substitution involves replacing variables in an expression with specific numerical values to evaluate the expression. In this context, substituting x = 63 and y = 57 into the special product formulas allows for the calculation of the resulting expressions. This technique is fundamental in algebra for simplifying and solving equations.