Factorizing Common Factors for a Perfect Square
- In algebra, a polynomial is a mathematical expression containing a combination of constants, variables and exponents that are added or subtracted from each other. There are different kinds of polynomials, usually identified by the number of terms in the expression. Monomials have one term: 4x^2. Binomials have two terms: 3x + 4. Trinomials have three terms: 6X^2 + 4X - 3.
- A perfect square trinomial is a special type of trinomial with certain tricks that you can use in order to factor it easily. They are formed by multiplying a binomial expression by itself. For example, X^2 + 10X + 25 is a perfect square trinomial because it can be factored as (X + 5)(X + 5) = (X + 5)^2.
- To determine whether a trinomial is a perfect square, take a look at each term. In a perfect square, the first and last terms are squares which means they are determined by a number multiplied by itself.
In X^2 + 10X + 25, the first term is X^2, the second 25. X^2 equals X x X which means it is a square and 25 equals 5 x 5 which is also a square. The middle term (10X) is determined from the square root of the first term (X) multiplied by the square root of the second term (5) then multiplied by 2. In this case, 5 x X x 2 = 10X. Thus, X^2 + 10X + 25 is a perfect square. - To factor a perfect square, you simply do the same thing as you would in order to recognize one. Take the square root of the first term. In X^2 + 10X + 25, the first term is X^2 and the square root of X^2 is X. Take the square root of the third term. Using the same example, the square root of 25 is 5. These two numbers are the first and second numbers of the binomial that the perfect square factors into.
To determine the sign for the binomial, look at the sign of the middle term. If the middle term is positive, the sign between the terms for the binomial is positive for both binomials. If the middle term is negative, one binomial has a plus sign, the other a negative sign.
X^2 + 10X + 25 is factored as (X + 5) (X + 5) or (X + 5)^2. If the perfect square is X^2 - 10X + 25, then the factors become X^2 - 10X + 25 = (X + 5) (X - 5). - Sometimes common factors can be factored out of a polynomial, reducing the trinomial left behind to a perfect square. In 4X^2 + 40X + 100, a 4 can be factored out of each term to reduce the polynomial to a perfect square: 4 (X^2 + 10X + 25) = 4 (X + 5) (X + 5) = 4 (X + 5)^2. Other times, a binomial or, possibly another trinomial, can be factored out to leave a perfect square behind which you can then factor the same way.
