A trinomial is a polynomial that has three terms. A polynomial has two or more algebraic terms. The first term has an exponent to the second degree (x^{2}), the second term is x, and the last term is a constant (just any number). Factoring trinomials is an extremely important algebra skill. Factoring polynomials can be tricky however with the detailed explanation below any problem can be solved.

**Trinomial **

**ax**^{2}** + bx + c**

**a = the x**^{2}** term **

**b = the x term **

**c = the constant value **

**Example 1:**

**a**^{2}** + 4a – 12**

First identify the values for a, b, and c.

a = 1

b = 4

c = -12

Remember when no number is in front of a variable, we assume the coefficient is 1.

Step 2: Find two numbers that when added together produce b and when multiplied together produce c. In this case we should list the factors of -12 first.

Factors of -12:

Possible Combinations | Does c = -12 when multiplied? | Does b = 4 when added? | Correct Combination |

4 x 3 | 12 | 7 | No |

4 x -3 | -12 | 1 | No |

-4 x -3 | 12 | -7 | No |

-4 x 3 | -12 | -1 | No |

6 x 2 | 12 | 8 | No |

-6 x 2 | -12 | -4 | No |

6 x -2 | -12 | 4 | Yes |

-6 x -2 | 12 | -8 | No |

12 x 1 | 12 | 13 | No |

-12 x 1 | -12 | -11 | No |

12 x -1 | -12 | 11 | No |

-12 x -1 | 12 | -13 | No |

The table above shows all the possible combinations for example 1. However only one combination equals both b and c (highlighted in bold). The correct combination of the two numbers should look like this:

(a – 2) (a + 6)

We can foil the problem to see if it matches the original problem.

**Example 2: **

Factor the following polynomial:

x^{2} + 2x -8

First identify the values of a, b, and c.

a = 1

b = 2

c =-8

Now consider the factors of c and when added together equal b.

Possible Combinations | Does c = -8 when multiplied? | Does b = 2 when added? | Correct Combination |

8 x 1 | 8 | 9 | No |

-8 x -1 | 8 | -9 | No |

8 x -1 | -8 | 7 | No |

-8 x 1 | 8 | -7 | No |

2 x 4 | 8 | 6 | No |

-2 x 4 | -8 | 2 | Yes |

2 x -4 | -8 | -2 | No |

-2 x -4 | 8 | -8 | No |

The correct combination should look like this:

(x – 2) (x + 4)

Use the foil method to check it against the original problem.