Topic: Special Products: Square of a Binomial

Essential Question

How can special products and factors help determine patterns from various real-life situations?

Introduction Man cannot live without a smoother relationship with others. So that when two persons are related to each other, their relationship can be described in two opposite ways. If Dr. Rubio is John’s teacher, then we can also say that John is Dr. Rubio’s student. This is the same true in Algebra, numbers/expressions too are related to each other. We can also say that 4 is related to 2 in manner that 4 is the square of 2 and 2 is the square root of 4.

Special Products

In mathematics products are obtained by multiplication. In this section, you will discover patterns that help you determine the products of polynomials. These are called special products. They are called special products because products are obtained through definite patterns.

Recall: Laws of Exponents1. The Product of Powers

am ∙ an = am+n

Examples:

x3 ∙ x2 = x5

x4 ∙ x5 = x9

Another Example

(2x3) (-3x4) =

-6x7

2. The Power of a Power

(am)n = amn

Examples:(x4)3 = x12

(x2)3 = x6

(4x3)2 = 16x6

Another Example

(3y4z)5 =

243y20z5

3. The Power of a Product

(ab)m = ambm

Examples:(2x)3 = 8x3

(2a2b4c7)4 = 16a8b16c28

Another Example

(-5x4y5z)2 =

25x8y10z2

Square of a Binomial

(x+y)2

(x-y)2

Multiply. We can find a shortcut.

(x + y) (x + y)

x² + xy + xy + y2

= x² + 2xy + y2Shortcut: Square the first term, add twicethe product of both terms and add the square of the second term.

This is a “Perfect Square Trinomial.”

(x + y)2

This is the square of a binomial pattern.

Multiply. Use the shortcut.

(4x + 5)2

= (4x)² + 2(4x●5) + (5)2

Shortcut:

= 16x² + 40x + 25

x² + 2xy + y2

Try these!

(x + 3)2

(5m + 8)2

(2x + 4y)2

(-4x + 7)2

x² + 6x + 9

25m² + 80m + 64

4x² + 16xy + 16y²

16x²- 56x + 49

Multiply. We can find a shortcut.

(x – y) (x – y)

x² - xy - xy + y2

= x² - 2xy + y2

This is a “Perfect Square Trinomial.”

(x – y)2

This is the square of a binomial pattern.

Multiply. Use the shortcut.

(3x - 7)2

Shortcut:

= 9x² - 42x + 49

x² - 2xy + y2

Try these!

(x – 7)2

(3p - 4)2

(4x - 6y)2

x² - 14x + 49

9p² - 24p + 16

16x² - 48xy + 36y²

Homework # 2