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Using Patterns to Multiply Two Binomials

We can use FOIL to find the product of any two binomials. Sometimes we can find certain binomial products more quickly by recognizing patterns.

For example, let’s first use FOIL to find (a + b)2.

Use the definition of exponential notation to write (a + b)2 as the product of two binomials. (a + b)2 = (a + b)(a + b)
Use FOIL.

Combine like terms.

Thus, (a + b)2 = a2 + 2ab + b2

  = a2 + ab + ba + b2

= a2 + 2ab + b2

Note:

Note that (a + b)2 is not equal to a2 + b2. Don’t forget the middle term, 2ab.

 

We obtain a similar pattern when we square a binomial that is a difference rather than a sum.

 

Formula — Square of a Binomial

Let a and b represent any real numbers.

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

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

 

When a binomial is squared, the resulting trinomial is called a perfect square trinomial.

For example, both a2 + 2ab + b2 and a2 - 2ab + b2 are perfect square trinomials.

 

Note:

When we refer to integers, a perfect square is an integer that is the square of another integer:

9 is a perfect square because it is the result of squaring 3.

A similar situation exists for variables:

a2 is a perfect square because it is the result of squaring a.

64n2 is a perfect square because it is the result of squaring 8n.

 

Example 1

Find: (6y2 + 5)2

Solution

The expression (6y2 + 5)2 is in the form (a + b)2.

So, we can use the formula for the square of a binomial.

(a + b)2

= a2 + 2ab + b2
Substitute 6y2 for a and 5 for b.

Simplify.

So, (6y2 + 5)2 = 36y4 + 60y2 + 25

(6y2 + 5)2

 

= (6y2)2 + 2(6y2)5 + (5)2

= 36y4 + 60y2 + 25

Note that (6y2 + 5)2 (6y2)2 + (5)2. Don’t forget the middle term, 60y2.

 

Example 2

Find: (3w - 7y)(3w - 7y)

Solution

Since (3w - 7y)(3w - 7y) = (3w - 7y)2, we can use the shortcut for the square of a binomial.

(a - b)2

= a2 - 2ab + b2
Substitute 3w for aand 7y for b.

Simplify.

(3w - 7y)2

 

= (3w)2 + 2(3w)(7y) + (7y)2

= 90w2 - 42wy + 49y2

So, (3w - 7y)(3w - 7y) = 90w2 - 42wy + 49y2.
Note that (3w2 - 7y)2 (3w2)2 - (7y)2. Don’t forget the middle term, -42wy2.
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