Special Binomial Products
Binomial
A binomial is polynomial with two terms
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example of a binomial
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And "Product" means the result you get after multiplying!
We are going to look at three special cases of multiplying binomials ...
Multiplying a Binomial by Itself
What happens when you square a binomial (in other words, multiply it by itself) .. ?
(a+b)2 = (a+b)(a+b) = ... ?
And what happens if you square of binomial with a minus inside?
(a-b)2 = (a-b)(a-b) = ... ?
And then there is one more special case... what if you multiply (a+b) by (a-b) ?
(a+b)(a-b) = ... ?
That was interesting! It ended up very simple. And it is called the "difference of two squares" (the two squares are a2 and b2).
The Three Cases
Here are the three results we just got:
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(a+b)2 = a2 + 2ab + b2 |
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(a-b)2 = a2 - 2ab + b2 |
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(a+b)(a-b) = a2 - b2 |
(called "the difference of squares") |
Remember those patterns, they will save you time and help you solve many algebra puzzles.
Using Them
So far we have just used "a" and "b", but they could be anything.
Example: (y+1)2
We can use the (a+b)2 case where "a" is y, and "b" is 1:
(y+1)2 = (y)2 + 2(y)(1) + (1)2 = y2 + 2y + 1
Example: (3x-4)2
We can use the (a-b)2 case where "a" is 3x, and "b" is 4:
(3x-4)2 = (3x)2 - 2(3x)(4) + (4)2 = 9x2 - 24x + 16
Example: (4y+2)(4y-2)
We know that the result will be the difference of two squares, because:
(a+b)(a-b) = a2 - b2
so:
(4y+2)(4y-2) = (4y)2 - (2)2 = 16y2 - 4
Sometimes you can recognize the pattern of the answer:
Example: can you work out which binomials to multiply to get 4x2 - 9
Hmmm... is that the difference of two squares?
Yes! 4x2 is (2x)2, and 9 is (3)2, so we have:
4x2 - 9 = (2x)2 - (3)2
And that can be produced by the difference of squares formula:
(a+b)(a-b) = a2 - b2
Like this ("a" is 2x, and "b" is 3):
(2x+3)(2x-3) = (2x)2 - (3)2 = 4x2 - 9
So the answer is that you can multiply (2x+3) and (2x-3) to get 4x2 - 9
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