Learn on PengienVision, Algebra 1Chapter 7: Polynomials and Factoring

Lesson 3: Multiplying Special Cases

In this Grade 11 enVision Algebra 1 lesson from Chapter 7, students learn to apply two special polynomial multiplication patterns: the square of a binomial, expressed as (a+b)² = a² + 2ab + b², and the difference of two squares, expressed as (a+b)(a-b) = a² - b². Students practice using these shortcuts to expand expressions like (5x-3)² and (5x+7)(5x-7), and also apply the patterns to mentally compute products of large numbers. The lesson builds conceptual understanding through visual models, numerical exploration, and a real-world pixel border application problem.

Section 1

Binomial Squares Pattern

Property

If $a$ and $b$ are real numbers,

(a+b)^2 = a^2 + 2ab + b^2

(a-b)^2 = a^2 - 2ab + b^2

This pattern is a shortcut for squaring a binomial.
To use it, you square the first term, square the last term, and then add or subtract double the product of the two terms.

Examples

To multiply $(z+6)^2$ , we use the pattern $(a+b)^2 = a^2 + 2ab + b^2$ . Here, $a=z$ and $b=6$ . The result is $(z)^2 + 2(z)(6) + (6)^2$ , which simplifies to $z^2 + 12z + 36$ .

To multiply $(3y+4)^2$ , we identify $a=3y$ and $b=4$ . Using the pattern, we get $(3y)^2 + 2(3y)(4) + (4)^2$ . This simplifies to $9y^2 + 24y + 16$ .

Section 2

Product of Conjugates Pattern

Property

A conjugate pair is two binomials of the form $(a-b)$ and $(a+b)$ . They have the same first term and the same last term, but one is a sum and the other is a difference.

If $a$ and $b$ are real numbers, the Product of Conjugates Pattern is:

(a-b)(a+b) = a^2 - b^2

The product is called a difference of squares.
To multiply conjugates, square the first term, square the last term, and write the result as a difference.

Examples

To multiply $(p-10)(p+10)$ , we recognize this as a product of conjugates where $a=p$ and $b=10$ . Using the pattern $a^2 - b^2$ , we get $(p)^2 - (10)^2$ , which simplifies to $p^2 - 100$ .

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Binomial Squares Pattern

Property

If $a$ and $b$ are real numbers,

(a+b)^2 = a^2 + 2ab + b^2

(a-b)^2 = a^2 - 2ab + b^2

This pattern is a shortcut for squaring a binomial.
To use it, you square the first term, square the last term, and then add or subtract double the product of the two terms.

Examples

To multiply $(z+6)^2$ , we use the pattern $(a+b)^2 = a^2 + 2ab + b^2$ . Here, $a=z$ and $b=6$ . The result is $(z)^2 + 2(z)(6) + (6)^2$ , which simplifies to $z^2 + 12z + 36$ .

To multiply $(3y+4)^2$ , we identify $a=3y$ and $b=4$ . Using the pattern, we get $(3y)^2 + 2(3y)(4) + (4)^2$ . This simplifies to $9y^2 + 24y + 16$ .

Section 2

Product of Conjugates Pattern

Property

A conjugate pair is two binomials of the form $(a-b)$ and $(a+b)$ . They have the same first term and the same last term, but one is a sum and the other is a difference.

If $a$ and $b$ are real numbers, the Product of Conjugates Pattern is:

(a-b)(a+b) = a^2 - b^2

The product is called a difference of squares.
To multiply conjugates, square the first term, square the last term, and write the result as a difference.

Examples

To multiply $(p-10)(p+10)$ , we recognize this as a product of conjugates where $a=p$ and $b=10$ . Using the pattern $a^2 - b^2$ , we get $(p)^2 - (10)^2$ , which simplifies to $p^2 - 100$ .

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Binomial Squares Pattern

Property

If $a$ and $b$ are real numbers,

(a+b)^2 = a^2 + 2ab + b^2

(a-b)^2 = a^2 - 2ab + b^2

This pattern is a shortcut for squaring a binomial.
To use it, you square the first term, square the last term, and then add or subtract double the product of the two terms.

Examples

To multiply $(z+6)^2$ , we use the pattern $(a+b)^2 = a^2 + 2ab + b^2$ . Here, $a=z$ and $b=6$ . The result is $(z)^2 + 2(z)(6) + (6)^2$ , which simplifies to $z^2 + 12z + 36$ .

To multiply $(3y+4)^2$ , we identify $a=3y$ and $b=4$ . Using the pattern, we get $(3y)^2 + 2(3y)(4) + (4)^2$ . This simplifies to $9y^2 + 24y + 16$ .

Section 2

Product of Conjugates Pattern

Property

A conjugate pair is two binomials of the form $(a-b)$ and $(a+b)$ . They have the same first term and the same last term, but one is a sum and the other is a difference.

If $a$ and $b$ are real numbers, the Product of Conjugates Pattern is:

(a-b)(a+b) = a^2 - b^2

The product is called a difference of squares.
To multiply conjugates, square the first term, square the last term, and write the result as a difference.

Examples

To multiply $(p-10)(p+10)$ , we recognize this as a product of conjugates where $a=p$ and $b=10$ . Using the pattern $a^2 - b^2$ , we get $(p)^2 - (10)^2$ , which simplifies to $p^2 - 100$ .