The Multiplication of Polynomials

The Multiplication of Polynomials

Student Outcomes

• Students develop the distributive property for application to polynomial multiplication.
• Students connect multiplication of polynomials with multiplication of multi-digit integers.

New York State Common Core Math Algebra II, Module 1, Lesson 2

Worksheets for Algebra 2

Classwork

Opening Exercise

Show that 28 × 27 = (20 + 8)(20 + 7) using an area model. What do the numbers you placed inside the four rectangular regions you drew represent?

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Example 1

Use the tabular method to multiply (𝑥 +8)(𝑥 + 7) and combine like terms.

Exercises 1–2

1. Use the tabular method to multiply (𝑥² + 3𝑥 + 1)(𝑥² − 5𝑥 + 2) and combine like terms.
2. Use the tabular method to multiply (𝑥² + 3𝑥 + 1)(𝑥² − 2) and combine like terms.

Example 2

Multiply the polynomials (𝑥 − 1)(𝑥⁴ + 𝑥³ + 𝑥² + 𝑥 + 1) using a table. Generalize the pattern that emerges by writing down an identity for (𝑥 − 1)(𝑥ⁿ + 𝑥^𝑛−1 + ⋯ + 𝑥² + 𝑥 + 1) for 𝑛 a positive integer.

Exercises 3–4

3. Multiply (𝑥 − 𝑦)(𝑥³ + 𝑥²𝑦 + 𝑥𝑦² + 𝑦³) using the distributive property and combine like terms. How is this calculation similar to Example 2?
4. Multiply (𝑥² − 𝑦²)(𝑥² + 𝑦²) using the distributive property and combine like terms. Generalize the pattern that emerges to write down an identity for (𝑥ⁿ − 𝑦ⁿ)(𝑥ⁿ + 𝑦ⁿ) for positive integers 𝑛.

Relevant Vocabulary

• EQUIVALENT POLYNOMIAL EXPRESSIONS: Two polynomial expressions in one variable are equivalent if, whenever a number is substituted into all instances of the variable symbol in both expressions, the numerical expressions created are equal.
• POLYNOMIAL IDENTITY: A polynomial identity is a statement that two polynomial expressions are equivalent. For example, (𝑥 + 3)² = 𝑥² + 6𝑥 + 9 for any real number 𝑥 is a polynomial identity.
• COEFFICIENT OF A MONOMIAL: The coefficient of a monomial is the value of the numerical expression found by substituting the number 1 into all the variable symbols in the monomial. The coefficient of 3𝑥² is 3, and the coefficient of the monomial (3𝑥𝑦𝑧) ⋅ 4 is 12.
• TERMS OF A POLYNOMIAL: When a polynomial is expressed as a monomial or a sum of monomials, each monomial in the sum is called a term of the polynomial.
• LIKE TERMS OF A POLYNOMIAL: Two terms of a polynomial that have the same variable symbols each raised to the same power are called like terms.
• STANDARD FORM OF A POLYNOMIAL IN ONE VARIABLE: A polynomial expression with one variable symbol, 𝑥, is in standard form if it is expressed as 𝑎_𝑛𝑥^𝑛 +𝑎_{𝑛 - 1}𝑥^{𝑛 - 1} + ⋯ + 𝑎₁𝑥 + 𝑎₀, where 𝑛 is a non-negative integer, and 𝑎₀, 𝑎₁, 𝑎₂ … , 𝑎_𝑛 are constant coefficients with 𝑎_𝑛 ≠ 0. A polynomial expression in 𝑥 that is in standard form is often just called a polynomial in 𝑥 or a polynomial. The degree of the polynomial in standard form is the highest degree of the terms in the polynomial, namely 𝑛. The term 𝑎_𝑛𝑥^𝑛 is called the leading term and 𝑎_𝑛 (thought of as a specific number) is called the leading coefficient. The constant term is the value of the numerical expression found by substituting 0 into all the variable symbols of the polynomial, namely 𝑎₀.

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