Binomial Squares Pattern

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

This is another one:                        (y7)2
Multiply(y7)by itself.                     (y7)(y7)
Using FOIL, we get:                         y27y7y+49
And combining like terms:                y214y+49
And one more:                                (2x+3)2
Multiply.                                         (2x+3)(2x+3)
Use FOIL:                                       4x2+6x+6x+9
Combine like terms.                         4x2+12x+9
Looking at these results. Do you notice any patterns?
How about the number of terms? In each example we squared a binomial and the result was a trinomial.
(a+b)2=____+____+____
Now look at the first term in each result. Where did it come from?

 

This figure has three columns. The first column contains the expression x plus 9, in parentheses, squared. Below this is the product of x plus 9 and x plus 9. Below this is x squared plus 9x plus 9x plus 81. Below this is x squared plus 18x plus 81. The second column contains the expression y minus 7, in parentheses, squared. Below this is the product of y minus 7 and y minus 7. Below this is y squared minus 7y minus 7y plus 49. Below this is the expression y squared minus 14y plus 49. The third column contains the expression 2x plus 3, in parentheses, squared. Below this is the product of 2x plus 3 and 2x plus 3. Below this is 4x squared plus 6x plus 6x plus 9. Below this is 4x squared plus 12x plus 9.

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+____

(ab)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

Binomial Squares Pattern

If  are real numbers,

(a+b)2=a2+2ab+b2
(ab)2=a22ab+b2

No Alt Text

 

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+2104+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

x plus 5, in parentheses, squared. Above the expression is the general formula a plus b, in parentheses, squared.
Square the first term. x squared plus blank plus blank. Above the expression is the general form a squared plus 2 a b plus b squared.
Square the last term. x squared plus blank plus 5 squared.
Double the product. x squared plus 2 times x times 5 plus 5 squared. Above this expression is the general formula a squared plus 2 times a times b plus b squared.
Simplify. x squared plus 10 x plus 25.

Example

 Multiply: (y−3)2.(y−3)2.

Solution

y minus 3, in parentheses, squared. Above the expression is the general formula a minus b, in parentheses, squared.
Square the first term. y squared minus blank plus blank. Above the expression is the general form a squared plus 2 a b plus b squared.
Square the last term. y squared minus blank plus 3 squared.
Double the product. y squared minus y times y times 3 plus 3 squared. Above this expression is the general formula a squared plus 2 times a times b plus b squared.
Simplify. y squared minus 6 y plus 9.

Example

Multiply: (4x+6)2.(4x+6)2.

Solution

4 x plus 6, in parentheses, squared. Above the expression is the general formula a plus b, in parentheses, squared.
Use the pattern. 4 x squared plus 2 times 4 x times 6 plus 6 squared. Above this expression is the general formula a squared plus 2 times a times b plus b squared.
Simplify. 16 x squared plus 48 x plus 36.

 

Example

Multiply: (2x−3y)2.(2x−3y)2.

Solution

contains 2 x minus 3 y, in parentheses, squared. Above the expression is the general formula a plus b, in parentheses, squared.
Use the pattern. 2 x squared minus 2 times 2 x times 3 y plus 3 y squared. Above this expression is the general formula a squared minus 2 times a times b plus b squared.
Simplify. 4 x squared minus 12 x y plus 9 y squared.

 

Example

Multiply: (4u3+1)2.(4u3+1)2.

Solution

4 u cubed plus 1, in parentheses, squared. Above the expression is the general formula a plus b, in parentheses, squared.
Use the pattern. 4 u cubed, in parentheses, squared, plus 2 times 4 u cubed times 1 plus 1 squared. Above this expression is the general formula a squared plus 2 times a times b plus b squared.
Simplify. 16 u to the sixth power plus 18 u cubed plus 1.

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