Mental Math

Mental Math

Student Outcomes

• Students perform arithmetic by using polynomial identities to describe numerical relationships.

New York State Common Core Math Algebra II, Module 1, Lesson 7

Worksheets for Algebra II, Module 1, Lesson 7

Classwork

Opening Exercise

a. How are these two equations related? b. Explain the relationship between the polynomial identities 𝑥² − 1 = (𝑥 + 1)(𝑥 − 1) and

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Exercises 1–3

1. Compute the following products using the identity 𝑥² − 𝑎² = (𝑥 − 𝑎)(𝑥 + 𝑎). Show your steps.
   a. 6 ⋅ 8
   b. 11 ⋅ 19
   c. 23 ⋅ 17
   d. 34 ⋅ 26
2. Find two additional factors of 2¹⁰⁰ − 1.
3. Show that 8³ −1 is divisible by 7.

Lesson Summary

Based on the work in this lesson, students can convert differences of squares into products (and vice versa) using

𝑥² − 𝑎² = (𝑥 − 𝑎)(𝑥 +𝑎).

If 𝑥, 𝑎, and 𝑛 are integers with (𝑥 − 𝑎) ≠ ±1 and 𝑛 > 1, then numbers of the form 𝑥𝑛 − 𝑎𝑛 are not prime because

𝑥ⁿ − 𝑎ⁿ = (𝑥 − 𝑎)(𝑥^n-1 + 𝑎𝑥^n-2 + 𝑎²𝑥^n-3 + ⋯ + 𝑎^n-2𝑥 + 𝑎^n-1).

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