To square a binomial, multiply the binomial by itself:
(a + b)^{2} = (a + b)(a + b)
(a + b)^{2} | = | (a + b)(a + b) | |
= | a ^{2} + ab + ba + b ^{2} | ||
= | a ^{2} + ab + ab + b ^{2} | ||
= | a ^{2} +2ab + b ^{2} |
The square of a binomial is always the sum of:
When a binomial is squared, the resulting trinomial is called a perfect square trinomial.
Examples:
(x + 5)^{2} = x
^{2} +2(x)(5) + 5^{2} = x
^{2} + 10x + 25
(100 - 1)^{2} = 100^{2} +2(100)(- 1) + (- 1)^{2} = 10000 - 200 + 1 = 9801
(2x - 3y)^{2} = (2x)^{2} +2(2x)(- 3y) + (- 3y)^{2} = 4x
^{2} -12xy + 9y
^{2}
When we multiply two polynomials that are the sum and difference of
the same
2
terms --
(x + 5)
and
(x - 5)
for example -- we get an
interesting result:
(a + b)(a - b) | = | a(a) + a(- b) + ba + b(- b) | |
= | a ^{2} - ab + ab - b ^{2} | ||
= | a ^{2} - b ^{2} |
The product of the sum and difference of the same two terms is always the difference of two squares; it is the first term squared minus the second term squared. Thus, this resulting binomial is called a difference of squares.
Examples:
(7 - 2)(7 + 2) = 7^{2} -2^{2} = 49 - 4 = 45
(x + 9)(x - 9) = x
^{2} -9^{2} = x
^{2} - 81
(2x - y)(2x + y) = (2x)^{2} - y
^{2} = 4x
^{2} - y
^{2}
(3x
^{2} -2)(3x
^{2} +2) = (3x
^{2})^{2} -2^{2} = 9x
^{4} - 4
(- y + 5x)(- y - 5x) = (- y)^{2} - (5x)^{2} = y
^{2} -15x
^{2}