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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 has a GCF of 1, but it can be written as the product of the factors and
Trinomials of the form can be factored by finding two numbers with a product of and a sum of The trinomial for example, can be factored using the numbers and because the product of those numbers is and their sum is The trinomial can be rewritten as the product of and
A trinomial of the form can be written in factored form as where and
Can every trinomial be factored as a product of binomials?
No. Some polynomials cannot be factored. These polynomials are said to be prime.
Given a trinomial in the form factor it.
Factor
We have a trinomial with leading coefficient and We need to find two numbers with a product of and a sum of In [link] , we list factors until we find a pair with the desired sum.
Factors of | Sum of Factors |
---|---|
14 | |
2 |
Now that we have identified and as and write the factored form as
Does the order of the factors matter?
No. Multiplication is commutative, so the order of the factors does not matter.
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 can be rewritten as using this process. We begin by rewriting the original expression as and then factor each portion of the expression to obtain We then pull out the GCF of to find the factored expression.
To factor a trinomial in the form by grouping, we find two numbers with a product of and a sum of We use these numbers to divide the term into the sum of two terms and factor each portion of the expression separately, then factor out the GCF of the entire expression.
Factor by grouping.
We have a trinomial with and First, determine We need to find two numbers with a product of and a sum of In [link] , we list factors until we find a pair with the desired sum.
Factors of | Sum of Factors |
---|---|
29 | |
13 | |
7 |
So and
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