### Learning Outcomes

- Identify and factor the greatest common factor of a polynomial.
- Factor a trinomial with leading coefficient 1.
- Factor by grouping.

When we studied fractions, we learned that the **greatest common factor** (GCF) of two numbers is the largest number that divides evenly into both numbers. For instance, [latex]4[/latex] is the GCF of [latex]16[/latex] and [latex]20[/latex] because it is the largest number that divides evenly into both [latex]16[/latex] and [latex]20[/latex]. The GCF of polynomials works the same way: [latex]4x[/latex] is the GCF of [latex]16x[/latex] and [latex]20{x}^{2}[/latex] because it is the largest polynomial that divides evenly into both [latex]16x[/latex] and [latex]20{x}^{2}[/latex].

When factoring a polynomial expression, our first step is to check for a GCF. Look for the GCF of the coefficients, and then look for the GCF of the variables.

### A General Note: Greatest Common Factor

The **greatest common factor** (GCF) of polynomials is the largest polynomial that divides evenly into the polynomials.

### How To: Given a polynomial expression, factor out the greatest common factor

- Identify the GCF of the coefficients.
- Identify the GCF of the variables.
- Combine to find the GCF of the expression.
- Determine what the GCF needs to be multiplied by to obtain each term in the expression.
- Write the factored expression as the product of the GCF and the sum of the terms we need to multiply by.

### Example: Factoring the Greatest Common Factor

Factor [latex]6{x}^{3}{y}^{3}+45{x}^{2}{y}^{2}+21xy[/latex].

### Try It

Factor [latex]x\left({b}^{2}-a\right)+6\left({b}^{2}-a\right)[/latex] by pulling out the GCF.

Watch this video to see more examples of how to factor the GCF from a trinomial.

## 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 [latex]{x}^{2}+5x+6[/latex] has a GCF of 1, but it can be written as the product of the factors [latex]\left(x+2\right)[/latex] and [latex]\left(x+3\right)[/latex].

Trinomials of the form [latex]{x}^{2}+bx+c[/latex] can be factored by finding two numbers with a product of [latex]c[/latex] and a sum of [latex]b[/latex]. The trinomial [latex]{x}^{2}+10x+16[/latex], for example, can be factored using the numbers [latex]2[/latex] and [latex]8[/latex] because the product of these numbers is [latex]16[/latex] and their sum is [latex]10[/latex]. The trinomial can be rewritten as the product of [latex]\left(x+2\right)[/latex] and [latex]\left(x+8\right)[/latex].

### A General Note: Factoring a Trinomial with Leading Coefficient 1

A trinomial of the form [latex]{x}^{2}+bx+c[/latex] can be written in factored form as [latex]\left(x+p\right)\left(x+q\right)[/latex] where [latex]pq=c[/latex] and [latex]p+q=b[/latex].

### Q & A

**Can every trinomial be factored as a product of binomials?**

*No. Some polynomials cannot be factored. These polynomials are said to be prime.*

### How To: Given a trinomial in the form [latex]{x}^{2}+bx+c[/latex], factor it

- List factors of [latex]c[/latex].
- Find [latex]p[/latex] and [latex]q[/latex], a pair of factors of [latex]c[/latex] with a sum of [latex]b[/latex].
- Write the factored expression [latex]\left(x+p\right)\left(x+q\right)[/latex].

### Example: Factoring a Trinomial with Leading Coefficient 1

Factor [latex]{x}^{2}+2x - 15[/latex].

### Q & A

**Does the order of the factors matter?**

*No. Multiplication is commutative, so the order of the factors does not matter.*

### Try It

Factor [latex]{x}^{2}-7x+6[/latex].

## 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 [latex]2{x}^{2}+5x+3[/latex] can be rewritten as [latex]\left(2x+3\right)\left(x+1\right)[/latex] using this process. We begin by rewriting the original expression as [latex]2{x}^{2}+2x+3x+3[/latex] and then factor each portion of the expression to obtain [latex]2x\left(x+1\right)+3\left(x+1\right)[/latex]. We then pull out the GCF of [latex]\left(x+1\right)[/latex] to find the factored expression.

### A General Note: Factoring by Grouping

To factor a trinomial of the form [latex]a{x}^{2}+bx+c[/latex] by grouping, we find two numbers with a product of [latex]ac[/latex] and a sum of [latex]b[/latex]. We use these numbers to divide the [latex]x[/latex] term into the sum of two terms and factor each portion of the expression separately then factor out the GCF of the entire expression.

### How To: Given a trinomial in the form [latex]a{x}^{2}+bx+c[/latex], factor by grouping

- List factors of [latex]ac[/latex].
- Find [latex]p[/latex] and [latex]q[/latex], a pair of factors of [latex]ac[/latex] with a sum of [latex]b[/latex].
- Rewrite the original expression as [latex]a{x}^{2}+px+qx+c[/latex].
- Pull out the GCF of [latex]a{x}^{2}+px[/latex].
- Pull out the GCF of [latex]qx+c[/latex].
- Factor out the GCF of the expression.

### Example: Factoring a Trinomial by Grouping

Factor [latex]5{x}^{2}+7x - 6[/latex] by grouping.

### Try It

Factor the following.

- [latex]2{x}^{2}+9x+9[/latex]
- [latex]6{x}^{2}+x - 1[/latex]

In the next video we show another example of how to factor a trinomial by grouping.