Factoring Polynomials using Algebra

The reverse process of multiplying polynomials is defined as factoring polynomials. Consider that when we factor a number, we are searching for prime factors that multiply together to give the number. When a polynomial is factorized, usually only the polynomials are broken down to have integer coefficients and constants. Simplest way for factoring is that a common factor for every term. So we can factor out the common factor in the polynomials.

For example,  8=4*2, or 16=4*4


Problems:


Factoring Polynomials using Algebra Tiles:

Example for Factoring Polynomials using Algebra Tiles 1:

Factorize the polynomial x3 – 5x2 – 12x + 36

Solution for Factoring Polynomials using Algebra Tiles 1:

Sum of the coefficients of terms: 1–5–12 + 36 = 18 ≠ 0. ∴ (x–1) is not a factor.

Sum of the coefficients of even degree terms = –5 + 36 = 31

Sum of the coefficients of odd degree terms = 1 – 12 = –11

Since they are not equal we guess that (x + 1) is also not a factor. Let us check whether x – 2

is a factor. By synthetic division method

                 2 | 1       -5     -12        +36

                    |

                    |          +2      -6          -36
                    ________________________

                      1        -3     -18     |    0      

                  _________________________



Since the remainder is 0, (x – 2) is a factor. To find other factors

                      x2 – 3x – 18 = x2 – 6x + 3x – 18

                                       = x (x–6) + 3 (x–6) = (x + 3) (x – 6)

                       Therefore, x3 – 5x2 – 12x + 36 = (x–2) (x–6) (x+3)


Sample problem

Example for Factoring Polynomials using Algebra Tiles 2:

Factorize 2x3 + x2 – 5x + 2

Solution for Factoring Polynomials using Algebra Tiles 2:

Since the sum of the coefficients of all the terms: 2 + 1 – 5 + 2 = 5 – 5 = 0

We guess that (x – 1) is a factor.

By synthetic division,

                              1 | 2         +2        -5        +2

                                |

                                |              2        +3        -2

                                 ________________________

                                     2         3         -2     |   0        

                                _________________________

                     Remainder is 0. Quotient is 2x2 + 3x – 2

To find other factors, factorize the quotient,

                   2x2 + 3x – 2 = 2x2 + 4x – x – 2

                                = 2x (x + 2) – 1 (x + 2) = (x + 2) (2x – 1)

                   ∴ 2x3 + x2 – 5x + 2 = (x – 1) (x + 2) (2x – 1)