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Multiplying and Factoring Polynomial Expressions

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Lesson Plans and Worksheets for Algebra I
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Common Core For Algebra I

Examples, solutions, and videos to help Algebra I students learn how to use the distributive property to multiply a monomial by a polynomial and understand that factoring reverses the multiplication process. Students use polynomial expressions as side lengths of polygons and find area by multiplying. Students recognize patterns and formulate shortcuts for writing the expanded form of binomials whose expanded form is a perfect square or the difference of perfect squares.

New York State Common Core Math Algebra I, Module 4, Lesson 1, Lesson 2
Download worksheets for Algebra I, Module 4, Lesson 1
Download worksheets for Algebra I, Module 4, Lesson 2

Lesson 1 Summary

Factoring is the reverse process of multiplication. When factoring, it is always helpful to look for a GCF that can be pulled out of the polynomial expression. For example, 3ab - 6a can be factored as 3a(b - 2).
Factor the difference of perfect squares a2 - b2 as (a - b)(a + b)
When squaring a binomial (a + b)2 = a2 + 2ab + b2

Opening Exercise
Write expressions for the areas of the two rectangles in the figures given below.
Example 1:
Jackson has given his friend a challenge:
The area of a rectangle, in square units, is represented by 3a2 + a for some real number a. Find the length and width of the rectangle.
How many possible answers are there for Jackson’s challenge to his friend? List the answer(s) you find.
Exercises 1–3
Factor each by factoring out the greatest common factor:
1. 10ab + 5a
2. 3g3h - 9g2 + 12h
3. 6x2y3 + 9xy4 + 18y2

Example 2: Multiply Two Binomials
For example, fill in the table to identify the partial products of (x + 2)(x + 5). Then, write the product of (x + 2)(x + 5) in standard form.
Example 3: The Difference of Squares
Find the product of (x + 2)(x - 2). Use the distributive property to distribute the first binomial over the second.

Exercise 4
Factor the following examples of the difference of perfect squares.
Exercises 5–7
Factor each of the following differences of squares completely.
Example 4: The Square of a Binomial
Square the following general examples to determine the general rule for squaring a binomial.
a. (a + b)2
b. (a - b)2
Exercises 8–9
Square the binomial

Lesson 2 Summary

Multiplying binomials is an application of the distributive property; each term in the first binomial is distributed over the terms of the second binomial.
The area model can be modified into a tabular form to model the multiplication of binomials (or other polynomials) that may involve negative terms.
When factoring trinomial expressions (or other polynomial expressions), it is useful to look for a GCF as your first step.
Do not forget to look for these special cases:
  • The square of a binomial
  • The product of the sum and difference of two expressions.

Example 1: Using a Table as an Aid
Use a table to assist in multiplying (x + 7)(x + 3).
Exercise 1
Use a table to aid in finding the product of (2x + 1)(x + 4).
Exercises 2–6
Multiply the following binomials; note that every binomial given in the problems below is a polynomial in one variable, 𝒙, with a degree of one. Write the answers in standard form, which in this case takes the form ax2 + bx + c, where a, b, and c are constants.

Exercises 7–10
Factor the following quadratic expressions
Example 3: Quadratic Expressions
a. First, factor out the GCF. (Remember: When you factor out a negative number, all the signs on the resulting factor change.)
b. Now look for ways to factor further. (Notice the quadratic expression factors.)

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