1.5 Factoring polynomials  (Page 2/6)

Factoring a trinomial with leading coefficient 1

Although we should always begin by looking for a GCF, pulling out the GCF is not the only way that polynomial expressions can be factored. The polynomial $x^2+5x+6$ has a GCF of 1, but it can be written as the product of the factors $(x+2)$ and $(x+3).$

Trinomials of the form $x^2+bx+c$ can be factored by finding two numbers with a product of $c$ and a sum of $b.$ The trinomial $x^2+10x+16,$ for example, can be factored using the numbers $2$ and $8$ because the product of those numbers is $16$ and their sum is $10.$ The trinomial can be rewritten as the product of $(x+2)$ and $(x+8).$

Factoring by grouping

Trinomials with leading coefficients other than 1 are slightly more complicated to factor. For these trinomials, we can factor by grouping    by dividing the x term into the sum of two terms, factoring each portion of the expression separately, and then factoring out the GCF of the entire expression. The trinomial $2x^2+5x+3$ can be rewritten as $(2x+3)(x+1)$ using this process. We begin by rewriting the original expression as $2x^2+2x+3x+3$ and then factor each portion of the expression to obtain $2x(x+1)+3(x+1).$ We then pull out the GCF of $(x+1)$ to find the factored expression.