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Factoring a polynomial is expressing the polynomial as a product of two or more factors; it is somewhat the reverse process of multiplying.

To factor polynomials, we generally make use of the following properties or identities; along with other more techniques.

Perfect Square Trinomial: $$a^2 + 2ab + b^2 = ( a + b )^ 2$$ $$a^2 - 2ab + b^2 = ( a - b )^2$$

**Additional Definitions:**

Factoring Trinomials of the form: $$x^2 + bx + c$$ A trinomial of the form is factorable over the integers, if there are two numbers p and q such that $$p * q = c \space\space and \space\space p + q = b$$ If two such numbers, p and q, exist, then the factored form of $$x^2 + bx + c = ( x + p)( x + q)$$

- If c < 0 then p and q have different signs.
- If c > 0 and b > 0, then p > 0 and q > 0.
- If c > 0 and b < 0, then p < 0 and q < 0.

Factoring Trinomials of the form: $$ax^2 + bx + c$$ A trinomial of the form is factorable over the integers, if there are two numbers p and q such that $$p * q = ac \space\space and \space\space p + q = b.$$

- Find two numbers, p and q, satisfying the two properties: $$p * q = a * c \space\space and \space\space q + p = b$$
- Write the trinomial $$ax^2 + bx + c$$ as a four term polynomial: $$ax^2 + px + qx + c$$
- Factor the four term polynomial using grouping: $$( ax^2 + px ) + ( qx + c )$$

Factoring Example #1$$x^2 - 2x - 8$$

First we need to find two numbers, p and q, whose product is -8 and whose sum is -2.

We know that in order for a product to be negative, then one number must be negative and one must be positive.

We get that p = 2 and q = -4.

Thus $$x^2 - 2x - 8 = ( x + 2)( x – 4 )$$ Factoring Example #2$$ 6x^2 - 17x + 12$$ Step 1: $$p * q = 6 * 12 \space\space and \space\space q + p = -17$$

Since b < 0 and c > 0, then p and q are both negative integers.

p | q | p*q = 6*12 = 72 | q+p = -17 |
---|---|---|---|

-1 | -72 | 72 | -73 |

-2 | -36 | 72 | -37 |

-3 | -24 | 72 | -27 |

-4 | -18 | 72 | -22 |

-6 | -12 | 72 | -17 |

-8 | -9 | 72 | -17 |

So, p = -8 and q = -9.

Step 2: $$6x^2 - 8x - 9x + 12$$ Step 3: $$= ( 6x^ 2 - 8x ) - ( 9x - 12 )$$ $$= 2x ( 3x - 4 ) - 3 ( 3x - 4 )$$ $$= ( 2x - 3 ) ( 3x - 4 )$$

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- Last Updated: Jul 7, 2023 12:00 PM
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