## Factoring a Quadratic Trinomial by Grouping

Another method for factoring these kinds of quadratic trinomials is called factoring by grouping. Factoring by grouping can be a bit more tedious and is often not worth the trouble if you can find the correct factors by some quick trial and error. However, it works quite well when the factors are not immediately obvious, such as when you have a very large number of candidate factors. When this happens, the trial and error method becomes very tedious.

Factoring by grouping is best demonstrated with a few examples.

**Example:**

Given: |
5x^{2} + 11x + 2 |

Find the product |
(5)(2) = 10 |

Think of two factors of 10 that add up to 11: |
1 and 10 |

Write the 11 |
5x^{2} + 1x + 10x + 2 |

Group the two pairs of terms: |
(5x^{2} + 1x) + (10x + 2) |

Remove common factors from each group: |
x(5x + 1) + 2(5x + 1) |

Notice that the two quantities in parentheses are now identical. That means we can factor out a common factor of (5 |
(5x + 1)(x + 2) |

Given: |
4 |

Find the product |
(4)(–15) = –60 |

Think of two factors of –60 that add up to 7: |
–5 and 12 |

Write the 7 |
4x^{2} – 5x + 12x – 15 |

Group the two pairs of terms: |
(4x^{2} – 5x) + (12x – 15) |

Remove common factors from each group: |
x(4x – 5) + 3(4x – 5) |

Notice that the two quantities in |
(4x – 5)(x + 3) |

#### The Procedure

Given a general quadratic trinomial *ax*^{2} + *bx* + *c*

1. Find the product *ac*.

2. Find two numbers *h* and *k* such that

*hk* = *ac *(*h* and *k* are factors of the product of the coefficient of *x*^{2} and the constant term)

AND

*h* + *k* = *b *(*h* and *k* add to give the coefficient of *x*)

3. Rewrite the quadratic as *ax*^{2} + *hx* + *kx* + *c*

4. Group the two pairs of terms that have common factors and simplify.

(*ax*^{2} + *hx*) + (*kx* + *c*)

*x*(*ax* + *h*) + (*kx* + *c*)

(note: because of the way you chose *h* and *k*, you will be able to factor a constant out of the second parentheses, leaving you with two identical expressions in parentheses as in the examples).

· Remember that this won’t work for all quadratic

trinomials, because not all quadratic trinomials can be factored into products

of binomials with integer coefficients. If you have a non-factorable trinomial,

you will not be able to do step 2 above.