The Multiplication of Polynomials


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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 II, Module 1, Lesson 2 (pdf)

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?

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 + 𝑥𝑛−1 + ⋯ + 𝑥2 + 𝑥 + 1) for 𝑛 a positive integer.




Exercises 3–4

  1. Multiply (𝑥 − 𝑦)(𝑥3 + 𝑥2𝑦 + 𝑥𝑦2 + 𝑦3) using the distributive property and combine like terms. How is this calculation similar to Example 2?
  2. 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 𝑎𝑛𝑥𝑛 +𝑎𝑛 - 1𝑥𝑛 - 1 + ⋯ + 𝑎1𝑥 + 𝑎0, where 𝑛 is a non-negative integer, and 𝑎0, 𝑎1, 𝑎2 … , 𝑎𝑛 are constant coefficients with 𝑎𝑛 ≠ 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 𝑎𝑛𝑥𝑛 is called the leading term and 𝑎𝑛 (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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