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 Saturday 3rd of August

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# Multiplying Special Binomials

## Product of a Sum and a Difference

If we multiply the sum a + b and the difference a - b by using FOIL, we get

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

The inner and outer products have a sum of 0. So the product of a sum and a difference of the same two terms is equal to the difference of two squares.

The Product of a Sum and a Difference

(a + b)(a - b) = a2 -  b2

Example 1

Product of a sum and a difference

Find each product

a) (x + 2)(x - 2)

b) (b + 7)(b - 7)

c) (3x - 5)(3x + 5)

Solution

a) (x + 2)(x - 2) = x2 - 4

b) (b + 7)(b - 7) = b2 - 49

c) (3x - 5)(3x + 5) = 9x2 - 25

## Higher Powers of Binomials

To find a power of a binomial that is higher thatn 2, we can use the rule for squaring a binomial along with the method of multiplying binomials using the distributive property. Finding the second or higher power of a binomial is called expandin the binomial because the result has more terms than the original.

Example 2

Higher powers of a binomial

Expand each binomial.

a) (x + 4)3

b) (y - 2)4

Solution

 a) (x + 4)3 = (x + 4)2(x + 4) = (x2 + 8x + 16)(x + 4) = (x2 + 8x + 16)x + (x2 + 8x + 16)4 = x3 + 8x2 + 16x + 4x2 + 32x + 64 = x3 + 12x2 + 48x + 64
 b) (y - 2)4 = (y - 2)2(y - 2)2 = (y2 - 4y + 4)(y2 - 4y + 4) = (y2 - 4y + 4)(y2) + (y2 - 4y + 4)(-4y) + (y2 - 4y + 4)(4) = y4 - 4y3 + 4y2 - 4y3 + 16y2 - 16y + 4y2 - 16y + 16 = y4 - 8y3 + 24y2 + 16