Overview
Trinomial squares are also known as perfect square trinomials, and are the squares of binomial expressions. They factor as (a + b)(a + b) or (a – b)(a – b) where a and b are real numbers. Forms such as (a + b)(a -b) are special products that are also called the difference of squares.
Trinomial Squares
When a squared binomial such as (a + b)(a + b) is multiplied using FOIL, the values of a and b follow a specific pattern. Recall that the first term is a2, the outside and inside terms are ab + ba, and the last term is b2. If a polynomial follows the form ax2 + bx + c, a and c are perfect squares, and the b coefficient is twice the sum of ac, it is a perfect trinomial square. Suppose the polynomial is 36x2 + 60x + 25. 36x2 is a perfect square of 6x, and 25 is a perfect square of 5. The inner and outer terms are 30x + 30x or 60x. That polynomial is (6x + 5)2.
Figure 1: The perfect square trinomial follows a specific pattern.
Negatives
What if the squared binomial is (a – b)(a – b)? When it is multiplied using FOIL, a2 is a perfect square, and so is b2. The sign of 2ab is negative, because it is the sum of two negative products. Suppose the polynomial is 100x2 – 80x + 16. The square root of 100x2 is 10x, and the square root of 16 is -4. The product of 10 and -4 is -40, and twice -40 is -80. That polynomial is (10x – 4)2.
Figure 2: An example when the middle term is negative.
Other Patterns
If the trinomial follows the form –ax2 +bx +c or ax2 – bx – c or ax2 + bx – c, it does not follow the squared trinomial pattern. The coefficient of a squared term cannot be negative even if the term is a perfect square. When a negative is multiplied by another negative, the product is positive. Similarly, if the constant c is not a perfect square, the trinomial does not follow the squared trinomial pattern.
Differences of Squares
The last special pattern to consider is (a + b)(a – b). Since multiplication is commutative, it is also the same as (a – b)(a + b). It is called the difference of squares. When (a + b)(a – b) is multiplied using FOIL, the first term is a2, and the last term is b2. The outside term is –ab and the inside term is ab, which adds up as zero. They cancel each other out. Suppose a polynomial is 144x2 + 81. The square root of 144x2 is 12x and the square root of 81 is 9. Following the pattern, the factoring is (12x + 9)(12x – 9).
Figure 3: An example of the form (a + b)(a – b).
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