Overview:
A perfect square trinomial is the square of a binomial. It follows a pattern when it is factored, so that the first and last terms are perfect squares of monomials and the middle term is twice their product. If the pattern does not fit for a particular trinomial, it is not a perfect square trinomial.
How Is A Binomial Squared?
In order to multiply two binomials (a + b)(c + d), the product is ac + bc + cb + bd, or in other words, ac + 2bc + bd. The process for squaring a binomial is the same. Suppose the binomial is (x + 3). Squaring that binomial, (in other words (x + 3)2), is the same thing as saying (x + 3)(x + 3). Following the pattern, multiplying x∙x, gives x2, while multiplying 3x and adding x3 equals 6x and multiplying 3∙3 equals 9. The entire equation is x2 + 6x + 9.
What if the Binomial Has a Minus Sign?
Suppose the binomial were (x-4) instead. Squaring the binomial (x-4) is the same thing as multiplying (x-4)(x-4). Following the FOIL pattern, (x-4)(x-4) is the same thing as x2 – 4x – x4 + 49. In other words, (x – 4)2 equals x2 -8x + 49. The process is similar as to (a-b)(c-d), with a product of ac – bc – cb + bd. Multiplying two negatives, -b ∙-d, changes the sign to a positive number.
What Is the Pattern for a Perfect Square Trinomial?
The square of a binomial (x +a) is the same thing as multiplying (x + a)(x + a), or x2 + 2ax + a2. The square of a binomial (x-a) is the same thing as multiplying (x-a)(x-a) or x2– 2ax + a2. Therefore, a trinomial that follows the pattern x2 ± 2ax + a2 is the square of a binomial.
Testing a Trinomial
Suppose the trinomial is 4x2 + 20x + 25. The square root of 4x2 is 2x and the square root of 25 is 5. The middle term 20x can be factored as 2∙2x∙5. It follows the pattern, and can be factored as 2x + 5, because (2x + 5)2 equals 4x2 + 20x + 25. Similarly, if the trinomial is x2 -16x + 64, the square root of x2 is x and the square root of 64 is 8. The number 64 also has a negative square root of -8, because -8 ∙-8 also equals 64. The middle term -16x can be factored as 2∙-8∙x. It also follows the pattern, and can be factored as x-8, because (x-8)2 equals x2 -16x + 64. Suppose the trinomial were 36x2 +23x + 81. Is it a perfect square trinomial? The square root of the monomial 36x2 is 6x and the square root of 81 is 9, but the middle term 23x is not equal to 2∙6x∙9, or 108x.
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