Dividing by 𝒙 − 𝒂 and by 𝒙 + 𝒂

Dividing by 𝒙 − 𝒂 and by 𝒙 + 𝒂

Student Outcomes

• Students work with polynomials with constant coefficients to prove polynomial identities.

New York State Common Core Math Algebra II, Module 1, Lesson 6

Worksheets for Algebra 2

Classwork

Opening Exercise

Find the following quotients, and write the quotient in standard form.

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Exercise 1

1. Use patterns to predict each quotient. Explain how you arrived at your prediction, and then test it by applying the reverse tabular method or long division.

Example 1 What is the quotient of (𝑥²−𝑎²)/(𝑥−𝑎)? Use the reverse tabular method or long division

Exercises 2–4 2. Work with your group to find the following quotients. 3. Predict without performing division whether or not the divisor will divide into the dividend without a remainder for the following problems. If so, find the quotient. Then check your answer. 4. a. Find the quotient for 𝑛 = 2, 3, 4, and 8
b. What patterns do you notice?
c. Use your work in part (a) to write an expression equivalent to

Lesson Summary

Based on the work in this lesson, it can be concluded that the following statements are true for all real values of 𝑥 and 𝑎:

𝑥² − 𝑎² = (𝑥 − 𝑎)(𝑥 + 𝑎)
𝑥³ − 𝑎³ = (𝑥 − 𝑎)(𝑥² + 𝑎𝑥 + 𝑎²)
𝑥³ + 𝑎³ = (𝑥 + 𝑎)(𝑥² − 𝑎𝑥 + 𝑎²),

and it seems that the following statement is also an identity for all real values of 𝑥 and 𝑎: 𝑥^{𝑛 − 1} = (𝑥 − 1)(𝑥^{𝑛 − 1} + 𝑥^{𝑛 − 2} + 𝑥^{𝑛 − 3} + ⋯ + 𝑥¹ + 1), for integers 𝑛 > 1.

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