1. Topic: Special Products: Square of a Binomial

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

3. 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.

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.

5. Recall: Laws of Exponents
The Product of Powers
am ∙ an = am+n
Examples:
x3 ∙ x2 = x5
x4 ∙ x5 = x9

6. Another Example
(2x3) (-3x4) =
-6x7

7. (x2)3 = x6 (4x3)2 = 16x6 (am)n = amn Examples: (x4)3 = x12
2. The Power of a Power
(am)n = amn
Examples:
(x4)3 = x12
(x2)3 = x6
(4x3)2 = 16x6

8. Another Example
(3y4z)5 =
243y20z5

9. (2a2b4c7)4 = 16a8b16c28 (ab)m = ambm Examples: (2x)3 = 8x3
3. The Power of a Product
(ab)m = ambm
Examples:
(2x)3 = 8x3
(2a2b4c7)4 = 16a8b16c28

10. Another Example
(-5x4y5z)2 = 25x8y10z2

11. Square of a Binomial
(x+y)2
(x-y)2

12. Multiply. We can find a shortcut.
(x + y)2
This is the square of a binomial pattern.
(x + y) (x + y) x² xy xy y2 + + +
Shortcut: Square the first term, add twice
the product of both terms and add the
square of the second term.
= x² + 2xy + y2
This is a “Perfect Square Trinomial.”

13. Multiply. Use the shortcut.
(4x + 5)2
x² + 2xy + y2
Shortcut:
= (4x)² + 2(4x●5) + (5)2
= 16x² + 40x + 25

14. Try these! (x + 3)2 x² + 6x + 9 25m² + 80m + 64 (5m + 8)2 (2x + 4y)2
4x² + 16xy + 16y²
(-4x + 7)2
16x²- 56x + 49

15. Multiply. We can find a shortcut.
(x – y)2
This is the square of a binomial pattern.
(x – y) (x – y) x² xy xy y2 - - +
= x² - 2xy + y2
This is a “Perfect Square Trinomial.”

16. Multiply. Use the shortcut.
(3x - 7)2
x² - 2xy + y2
Shortcut:
= 9x² - 42x + 49

17. Try these! (x – 7)2 x² - 14x + 49 9p² - 24p + 16 (3p - 4)2 (4x - 6y)2
16x² - 48xy + 36y²

18. Homework # 2