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Lesson Plans and Worksheets for Algebra II

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More Lessons for Algebra

Common Core For Algebra

Student Outcomes

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

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

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

Example 2

Multiply the polynomials (𝑥 − 1)(𝑥^{4} + 𝑥^{3} + 𝑥^{2} + 𝑥 + 1) using a table. Generalize the pattern that emerges by writing
down an identity for (𝑥 − 1)(𝑥^{n} + 𝑥^{𝑛−1} + ⋯ + 𝑥^{2} + 𝑥 + 1) for 𝑛 a positive integer.

Exercises 3–4

- Multiply (𝑥 − 𝑦)(𝑥
^{3}+ 𝑥^{2}𝑦 + 𝑥𝑦^{2}+ 𝑦^{3}) using the distributive property and combine like terms. How is this calculation similar to Example 2? - Multiply (𝑥
^{2}− 𝑦^{2})(𝑥^{2}+ 𝑦^{2}) using the distributive property and combine like terms. Generalize the pattern that emerges to write down an identity for (𝑥^{n}− 𝑦^{n})(𝑥^{n}+ 𝑦^{n}) 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}= 𝑥^{2}+ 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 3𝑥
^{2}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
_{𝑛}𝑥^{𝑛}+𝑎_{𝑛 - 1}𝑥^{𝑛 - 1}+ ⋯ + 𝑎_{1}𝑥 + 𝑎_{0}, where 𝑛 is a non-negative integer, and 𝑎_{0}, 𝑎_{1}, 𝑎_{2}… , 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_{𝑛}𝑥^{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 𝑎_{0}.

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