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9.4
Special Cases
9.4 – Special Cases
Goals / “I can…”I can find the square of a binomialI 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 + 4However, it really means
(x + 2)(x + 2)so you have to FOIL it.
2
2
Square of a Binomial
Square the first term:
2nm
2m 2mn
2n22 2 nmnm
• Square the last term:
• Double the product of both terms:
2) Multiply: (3x + 2y)2
using (a + b)2 = a2 + 2ab + b2
(3x + 2y)2
a = 3x and b = 2y
Plug into the formulaa2 + 2ab + b2
(3x)2 + 2(3x)(2y) + (2y)2Simplify
9x2 + 12xy +4y2
Example #1
243 ba
29a ab24 216b
Example #2
22 sr
24r rs4 2s
Multiply (2a + 3)2
1. 4a2 – 9
2. 4a2 + 9
3. 4a2 + 36a + 9
4. 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
1. x2 + 2xy + y2
2. x2 – 2xy + y2
3. x2 + y2
4. x2 – y2
9.4 – Special Cases
Difference of Squares: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:
nmnm 2m
• Square the last term:2n
• Write the difference of the two squares:
22 nm
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
25
55 zz
2z
Multiply (4m – 3n)(4m + 3n)
1. 16m2 – 9n2
2. 16m2 + 9n2
3. 16m2 – 24mn - 9n2
4. 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.