# Factoring Trinomials of the Form ax^2 + bx + c; Perfect Squares

## The polynomial $ax^2 + bx + c$ can be factored using a variety of methods, including trial and error.

#### Key Points

• Some trinomials, known as perfect square trinomials, can be factored into two equal binomials.

• We can factor $a^2 − b^2$, the difference of two squares, by finding the terms that produce the perfect squares and substituting these quantities into the factorization form. When using real numbers, there is no factored form for the sum of two squares.

• Perfect square trinomials factor as the square of a binomial. To recognize them, look for whether (1) the first and last terms are perfect squares, and (2) the middle term is divisible by 2, and when halved, equals the product of the terms that when squared produce the first and last terms.

#### Terms

• A polynomial expression consisting of three terms, or monomials, separated by addition and/or subtraction symbols.

• A polynomial consisting of two terms, or monomials, separated by addition or subtraction symbols.

#### Figures

1. ##### Factorization

The polynomial $x^2+cx+d$ is factored to the form $(x+a)(x+b)$. This is based on the relationships that $a+b=c$ and $a*b=d$.

2. ##### Interactive Graph: Polynomial

Graph of a trinomial with the example equation $y=x^2-x-2$. Notice what happens when the signs are flipped from negative to positive.

## Factoring Trinomials

The polynomial $ax^2+bx+c$ can be factored using a variety of methods. One such method is trial and error (Figure 2).

Ultimately, the trinomial should be factored in the form $(px+q)(rx+s)$, where p, q, r, and s are constants, and x is a variable. Using trial and error, we can find values for each of the constants, using the FOIL method to determine whether the constants used produce the trinomial $ax^2+bx+c$. As seen in (Figure 1), FOIL stands for the order in which one multiplies the four terms - first, outside, inside, last. The order isn't important; however, the acronym is useful to prevent missing a multiplication term or accidentally multiplying the same terms twice.  We know that the product of px and rx must equal ax2. Additionally, the sum of products $px \ast s$ and $q \ast rx$ must equal bx. Finally, the product of q and s must equal c.

## Perfect Squares

Some trinomials, known as perfect square trinomials, can be factored into two equal binomials. For example:

$a^2+2ab+b^2=(a+b)^2$

and

$a^2-2ab+b^2=(a-b)^2$

Perfect square trinomials always factor as the square of a binomial.

To recognize a perfect square trinomial, look for the following features:

1. The first and last terms are perfect squares.
2. The middle term is divisible by 2.

For example, $x^2-10x+25$ can be identified as a perfect square because x2 is the square of x, and 25 is the square of 5. The middle term (-10x) is divisible by 2 (equalling -5x).

Given that the coefficient of x2 is 1, we know that the factored form will be $(x+a)(x+b)$, where a and b are to-be-determined coefficients. We need $x \ast b+a \ast x$ to equal -10x, and a*b to equal 25. Filling in -5 for a and b, we find a plausible solution that reads (x-5)(x-5), or (x-5)2. This is a perfect square.

#### Key Term Glossary

binomial
A polynomial consisting of two terms, or monomials, separated by addition or subtraction symbols.
##### Appears in these related concepts:
coefficient
a constant by which an algebraic term is multiplied.
##### Appears in these related concepts:
constant
An identifier that is bound to an invariant value.
##### Appears in these related concepts:
factor
To find all the factors of (a number or other mathematical object) (the objects that divide it evenly).
##### Appears in these related concepts:
polynomial
an expression consisting of a sum of a finite number of terms, each term being the product of a constant coefficient and one or more variables raised to a non-negative integer power, such as $a_n x^n + a_{n-1}x^{n-1} + ... + a_0 x^0$. Importantly, because all exponents are positive, it is impossible to divide by x.
##### Appears in these related concepts:
square
The second power of a number, value, term or expression.
##### Appears in these related concepts:
term
any value (variable or constant) or expression separated from another term by a space or an appropriate character, in an overall expression or table.
##### Appears in these related concepts:
trinomial
A polynomial expression consisting of three terms, or monomials, separated by addition and/or subtraction symbols.
##### Appears in these related concepts:
variable
A symbol that represents a quantity in a mathematical expression, as used in many sciences