Objective The student will be able to: multiply special binomials.

There are formulas (shortcuts) that work for certain polynomial multiplication problems. (a + b) 2 = a 2 + 2ab + b 2 (a - b) 2 = a 2 – 2ab + b 2 (a - b)(a + b) = a 2 - b 2 Being able to use these formulas will help you in the future when you have to factor. If you do not remember the formulas, you can always multiply using distributive, FOIL, or the box method.

Let’s try one! 1) Multiply: (x - 7) 2 You can multiply this by rewriting this as (x - 7)(x - 7) OR You can use the following rule as a shortcut: (a + b) 2 = a 2 + 2ab + b 2 For comparison, I’ll show you both ways.

1) Multiply (x -7)(x - 7) First terms: Outer terms: Inner terms: Last terms: Combine like terms. x 2 -14x + 49 x-7 x x 2 -7x +49 Now let’s do it with the shortcut! x2x2 -7x +49 Notice you have two of the same answer?

1) Multiply: (x - 7) 2 using (a - b) 2 = a 2 - 2ab + b 2 a is the first term, b is the second term (x -7) 2 a = x and b = -7 Plug into the formula a 2 - 2ab + b 2 (x) 2 - 2(x)(7) + (7) 2 Simplify. x 2 - 7x+ 49 This is the same answer! That’s why the 2 is in the formula!

2) Multiply (x + 9)(x - 9) First terms: Outer terms: Inner terms: Last terms: Combine like terms. x 2 – 81 x9 x -9 x 2 -9x 9x -81 This is called the difference of squares. x2x2 +9x -9x -81 Notice the middle terms eliminate each other!

2) Multiply (x – 9)(x + 9) using (a – b)(a + b) = a 2 – b 2 You can only use this rule when the binomials are exactly the same except for the sign. (x – 9)(x + 9) a = x and b = 9 (x) 2 – (9) 2 x 2 – 81

Example 3

Example 4

Example 5 (3a – 5)(3a + 5) Product of sum and difference!

Example 6

Example 7 Find the area of the large square, and the area of the small square. Subtract the smaller area from the larger area… your solution will be the area of the shaded region!

Multiply: (3x + 2y) 2 using (a + b) 2 = a 2 + 2ab + b 2 (3x + 2y) 2 a = 3x and b = 2y Plug into the formula a 2 + 2ab + b 2 (3x)2 + 2(3x)(2y) + (2y)2 Simplify 9x xy +4y 2

Multiply (2a + 3) 2 1.4a 2 – 9 2.4a a a a a + 9

Multiply: (x – 5) 2 using (a – b) 2 = a 2 – 2ab + b 2 Everything is the same except the signs! (x) 2 – 2(x)(5) + (5) 2 x 2 – 10x ) Multiply: (4x – y) 2 (4x) 2 – 2(4x)(y) + (y) 2 16x 2 – 8xy + y 2

Multiply (x – y) 2 1.x 2 + 2xy + y 2 2.x 2 – 2xy + y 2 3.x 2 + y 2 4.x 2 – y 2

Multiply: (y – 2)(y + 2) (y) 2 – (2) 2 y 2 – 4 Multiply: (5a + 6b)(5a – 6b) (5a) 2 – (6b) 2 25a 2 – 36b 2

Multiply (4m – 3n)(4m + 3n) 1.16m 2 – 9n m 2 + 9n m 2 – 24mn - 9n m mn + 9n 2

More practice!

Even more practice! Find the area of the shaded region.. Why does this matter?

A few more…

Exit ticket