9.4 Special Cases

9.4 – Special Cases Goals / “I can…” I can find the square of a binomial I can find the difference of squares

9.4 – Special Cases IMPORTANT: If you don’t want to take the rules and store them in your brain to recall at any time, you can always FOIL the binomials like yesterday.

9.4 – Special Cases There are certain binomial combinations that occur frequently. They form patterns when multiplied. If you remember the pattern, you can quickly simplify.

9.4 – Special Cases Binomial Square: (x + 2) Most people get this wrong because they distribute the square and get x + 4x + 4 However, it really means (x + 2)(x + 2) so you have to FOIL it. 2 2

Square the first term: Square of a Binomial Double the product of both terms: Square the last term:

2) Multiply: (3x + 2y)2 using (a + b)2 = a2 + 2ab + b2 a = 3x and b = 2y Plug into the formula a2 + 2ab + b2 (3x)2 + 2(3x)(2y) + (2y)2 Simplify 9x2 + 12xy +4y2

Example #1

Example #2

Multiply (2a + 3)2 4a2 – 9 4a2 + 9 4a2 + 36a + 9 4a2 + 12a + 9

Multiply: (x – 5)2 using (a – b)2 = a2 – 2ab + b2 Everything is the same except the signs! (x)2 – 2(x)(5) + (5)2 x2 – 10x + 25

4) Multiply: (4x – y)2 (4x)2 – 2(4x)(y) + (y)2 16x2 – 8xy + y2

Multiply (x – y)2 x2 + 2xy + y2 x2 – 2xy + y2 x2 + y2 x2 – y2

9.4 – Special Cases Difference of Squares: (2x + 3)(2x – 3) notice they are the same except for the sign

Product of the Sum & Difference Square the first term: Square the last term: Write the difference of the two squares:

9.4 – Special Cases (2x + 3)(2x – 3) = 4x – 6x + 6x – 9 = 4x – 9 2 2

5) Multiply (x – 3)(x + 3) using (a – b)(a + b) = a2 – b2 You can only use this rule when the binomials are exactly the same except for the sign. (x – 3)(x + 3) a = x and b = 3 (x)2 – (3)2 x2 – 9

Example #2

Multiply (4m – 3n)(4m + 3n) 16m2 – 9n2 16m2 + 9n2 16m2 – 24mn - 9n2

9.4 – Special Cases (x + 7) 2

9.4 – Special Cases (x – 9) 2

There are formulas (shortcuts) that work for certain polynomial multiplication problems. (a + b)2 = a2 + 2ab + b2 (a - b)2 = a2 – 2ab + b2 (a - b)(a + b) = a2 - b2 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.