Binomial Squares Pattern – Most mathematicians love to search for patterns that will make their work less difficult. A very good example of this is squaring binomials. While you can always get the product by writing the binomial twice and using the methods of the last section, there is not much work to do if you learn to use a pattern.
Let’s start by looking at(x+9)2.
What does this mean? (x+9)2
It means to multiply(x+9)by itself. (x+9)(x+9)
Then, using FOIL, we get: x2+9x+9x+81
Combining like terms gives: x2+18x+81
The first term is the product of the first terms of each binomial. Since the binomials are the same, it is just the square of the first term!
(a+b)2=a2+____+____
To get the first term of the product, square the first term.
Where did the last term originate from? Look at the examples and find the pattern.
The last term is the product of the last terms, which is the square of the last term.
(a+b)2=____+____+b2
To get the last term of the product, square the last term.
Finally, look at the middle term. Notice it came from adding the “outer” and the “inner” terms—which are both the same! So the middle term is twice the product of the two terms of the binomial.
(a+b)2=____+2ab+____
(a−b)2=____−2ab+____
To get the middle term of the product, multiply the terms and double their product.
To put it all together
Binomial Squares Pattern
If are real numbers,
To square a binomial:
- square the first term
- square the last term
- double their product
A number of examples will help to verify the pattern.
(10+4)2
Square the first term. 102+___+___
Square the last term. 102+___+42
Double their product. 102+2⋅10⋅4+42
Simplify. 100+80+16
Simplify. 196
To multiply (10+4)2 you’d follow the Order of Operations.
(10+4)2
(14)2
196
And the pattern sure works!
Example
Multiply: (x+5)2.
Solution
Square the first term. | |
Square the last term. | |
Double the product. | |
Simplify. |
Example
Solution
Square the first term. | |
Square the last term. | |
Double the product. | |
Simplify. |
Example
Solution
Use the pattern. | |
Simplify. |
Example
Multiply: (2x−3y)2.(2x−3y)2.
Solution
Use the pattern. | |
Simplify. |
Example
Multiply: (4u3+1)2.(4u3+1)2.
Solution
Use the pattern. | |
Simplify. |