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A binomial is a polynomial consisting of two terms. When squaring a binomial such as x + 5, start by
writing the binomial next to itself. It is important to remember to multiply each term in the first binomial
by each term in the second. You may remember this as FOIL: the process of multiplying first, outer, inner,
and last terms. For example, (x + 5)2 = (x + 5) (x + 5) so
firsts
outers
inners
lasts
®
®
®
®
x ×x = x 2
x ×5 = 5x
5 × x = 5x
5 ×5 = 25
Adding these terms yields x 2 + 5x + 5x + 25. Note that the middle two terms will always be like terms so
you can combine them: x 2 + 10x + 25.
Example 1: Square the binomial (80 + h).
Solution:
(80 + h) 2 =
(80 + h) (80 + h) =
6400 + 80h + 80h + h 2 =
6400 + 160h + h 2 =
2
h + 160h + 6400
Use the definition of “square”.
Multiply each term in first binomial by each term in the
second using FOIL.
Combine like terms: 80h + 80h = 160h
You may wish to rewrite the expression so that the exponents
are in descending order.
The easiest way to cube a binomial is to start by squaring the binomial using FOIL (as done above) and to
then multiply the answer by the leftover binomial. The cube of (x + 5) is written as (x + 5) 3, and it can be
rewritten as (x + 5)2 (x + 5) . We know from the above discussion that (x + 5)2 = x2 +10x + 25, so we can
easily substitute the result into the expression: (x + 5) 3 = (x + 5) 2 (x + 5) = (x2 + 10x + 25) (x + 5).
Now you have only to multiply. It might be easier to understand (and avoid mistakes!) if you distribute the
binomial as follows: x (x2 + 10x + 25) + 5(x2 + 10x + 25).
Next you must distribute the x and the 5 and finish the problem by combining like terms:
x (x2 +10x + 25) + 5(x2 + 10x + 25) =
x3 + 10x2 + 25x + 5x2 + 50x + 125 =
3
2
x + 15x + 75x + 125 =
Split up the binomial term (x + 5).
Distribute the x and the 5 throughout.
Rewrite the expression combining like terms.
Example 2: Cube the binomial (x – 2).
Solution:
(x – 2) 3 =
(x – 2) 2 (x – 2)
(x – 2) 2 =
(x – 2)(x –2) =
x2 – 2 x – 2 x + 4 =
x2 – 4 x + 4 =
Now substitute this result into (x – 2) 2 (x – 2):
(x2 – 4x + 4)(x – 2) =
x (x2 – 4x + 4) + –2(x2 – 4x + 4) =
Rewrite the cubed binomial as the product of a squared
binomial and 1st-degree binomial.
Use the definition of “square”.
Multiply each term in one expression by each term in the
other with FOIL.
Combine like terms.
Substitute result for binomial square
Distribute the binomial term x – 2.
·
Caution! Be very careful with the negative -- if you don’t trust yourself to distribute the subtraction
throughout the parentheses, rewrite the subtraction as adding a negative as done here: x – 2 = x + -2
Distribute the x and -2 throughout the parentheses.
Combine like terms.
x3 – 4x2 + 4x + -2x2 + 8x – 8 =
x3 – 6x2 + 12x – 8
Example 3: Expand (80 + h)3.
Solution:
We know from Example 1 that (80 + h) 2 = h2 + 160h + 6400. Using this information, we have:
(80 + h) 3 =
(80 + h) 2 (80 + h) =
(h
2
Rewrite cube.
Substitute for (80 + h)2.
+ 160h + 6400) (80 + h) =
80(h 2 + 160 h + 6400) + h(h 2 + 160 h + 6400) =
2
3
2
80h + 12,800h + 512,000 + h + 160h + 6400h =
2
3
240h + 19,200h + 512,000 + h =
3
2
h +240h + 19,200h + 512,000
Split up the 80 and the h.
Distribute the 80 and the h throughout.
Combine like terms.
If desired, rewrite the expression in
descending order.
***Note that it is also correct to write the terms in ascending order: 512,000 + 19,200h + 240h 2 + h 3.