Factoring Polynomials

Definition:  Factoring is the process of writing a polynomial as a product of factors.

Types of Factoring:

1.  The first step in factoring is to find the CF [common factor].

Examples:                     a)\, \, 3x+6=3\left ( x+2 \right )

                                        b)\, \, 2x^{3}-6x=2x\left ( x^{2}-3 \right )

                                        c)\, \, \left ( x+3 \right )^{2}-4\left ( x+3 \right )=\left ( x+3 \right )\left [ x+3-4 \right ]=\left ( x+3 \right )\left ( x-1 \right )

                                        d)\, \, \frac{1}{2}x^{3}+2x^{2}-5x=\frac{1}{2}x^{3}+\frac{4}{2}x^{2}-\frac{10}{2}x=\frac{1}{2}x\left ( x^{2}+4x-10 \right )

in example d) above, one is trying to find the common factor so the remaining factors have integer coefficients.

2.  Difference of Two Perfect Squares –  a^{2}-b^{2}=\left ( a+b \right )\left ( a-b \right )

Examples:                     a)\, \, 16y^{2}-9=\left ( 4y+3 \right )\left ( 4y-3 \right )

                                        b)\, \, 16x^{2}-\frac{1}{9}=\left ( 4x+\frac{1}{3} \right )\left ( 4x-\frac{1}{3} \right )

                                         c)\, \, \left ( x-1 \right )^{2}-4=\left ( x-1+2 \right )\left ( x-1-2 \right )=\left ( x+1 \right )\left ( x-3 \right )

                                         d)\, \, 9u^{2}-4v^{2}=\left ( 3u+2v \right )\left ( 3u-2v \right )

3.  Perfect Square Trinomials –  a^{2}+2ab+b^{2}=\left ( a+b \right )\left ( a+b \right )=\left ( a+b \right )^{2}

                                                      a^{2}-2ab+b^{2}=\left ( a-b \right )\left ( a-b \right )=\left ( a-b \right )^{2}

Examples:                        a)\, \, x^{2}-4x+4=\left ( x-2 \right )\left ( x-2 \right )=\left ( x-2 \right )^{2}

                                           b)\, \, 9u^{2}+24uv+16v^{2}=\left ( 3u+4v \right )\left ( 3u+4v \right )=\left ( 3u+4v \right )^{2}

                                           c)\, \, x^{2}-\frac{4}{3}x+\frac{4}{9}=\left ( x-\frac{2}{3} \right )\left ( x-\frac{2}{3} \right )=\left ( x-\frac{2}{3} \right )^{2}

4.  Sum or Difference of Two Cubes — a^{3}+b^{3}=\left ( a+b \right )\left ( a^{2}-ab+b^{2} \right )

                                                                 a^{3}-b^{3}=\left ( a-b \right )\left ( a^{2}+ab+b^{2} \right )

Examples:                          a)\, \, x^{3}-8=\left ( x-2 \right )\left ( x^{2} +2x+4\right )

                                             b)\, \, y^{3}+64=\left ( y+4 \right )\left ( y^{2} -4y+16\right )

                                             c)\, \, 8t^{3}-1=\left ( 2t-1 \right )\left ( 4t^{2} +2t+1\right ) 

                                             d)\, \, u^{3}+27v^{3}=\left ( u+2v \right )\left ( u^{2} -3uv+9v^{2}\right )

5.  Trinomials with Binomial Factors — ax^{2}+bx+c=\left ( \square x+\square \right )\left ( \square x+\square \right )

When the LC is 1, that is a=1, and c is positive:  x^{2}+bx+c=\left ( x+\square \right )\left ( x+\square \right )

                                                                                   x^{2}-bx+c=\left ( x-\square \right )\left ( x-\square \right ) 

Examples:                    a)\, \, s^{2}-5s+6=\left ( s-2 \right )\left ( s-3 \right )

                                       b)\, \, x^{2}+3x+2=\left ( x+2 \right )\left ( x+1 \right )

When the LC is not equal to 1, and c is positive:  ax^{2}\pm bx+c=\left ( \square x\pm \square \right )\left ( x\pm \square \right )

Examples:                    a)\, \, 3x^{2}-5x+2=\left ( 3x-2 \right )\left ( x-1 \right )      

       Check:    F\rightarrow 3x*x=3x^{2},\; O+I\rightarrow -3x+\left ( -2x \right )=-5x,\; L\rightarrow -2\left ( -1 \right )=+2

                                       b)\, \, 5x^{2}+26x+5=\left ( 5x+1 \right )\left ( x+5 \right )

       Check:    F\rightarrow 5x*x=5x^{2},\; O+I\rightarrow \left ( 5x*5=25x \right )+\left ( 1x\right )=26x,\; L\rightarrow 1\left ( 5 \right )=+5

                                       c)\, \, 3x^{2}+10x+8=\left ( 3x+4 \right )\left ( x+2 \right )

       Check:   F\rightarrow 3x*x=3x^{2},\; O+I\rightarrow \left ( 3x*2=6x \right )+\left ( 4x\right )=10x,\; L\rightarrow 4\left ( 2 \right )=+8

                                       d)\, \, 15x^{2}-11x+2=\left ( 3x-1 \right )\left ( 5x-2 \right )

       Check:   F\rightarrow 3x*5x=15x^{2} 

                      O+I\rightarrow \left [ 3x\left ( -2 \right )=-6x \right ]+\left [ -1\left ( 5x \right )=-5x\right ]=-11x

                       L\rightarrow \left ( -1 \right )\left ( -2 \right )=+2

Always check F – first, O – outer, I – inner, L – Last.  F multiplied together = First term of the trinomial, O multiplied together plus I multiplied together = the middle term of the trinomial, and L multiplied together = the last term of the trinomial.

When the LC is 1, that is a=1, and c is negative:  x^{2}\pm bx-c=\left ( x\pm \square \right )\left ( x\mp \square \right ) 

Again, the O+I totals the middle term of the trinomial.  When c is negative, one of the two blanks will be a positive number and the other a negative number.  If b is postive the total of O+I will be positive and if b is negative the total of the O+I will be negative.

Examples:             a)\, \, x^{2}+x-2=\left ( x+2 \right )\left ( x-1 \right )

                                b)\, \, x^{2}-x-2=\left ( x-2 \right )\left ( x+1 \right )

When a is not equal to 1 and c is negative, the same principles work.  The O+I must result in the value of b and one factor of c is positive and one is negative.  Remember to check.

Examples:             a)\, \, 3x^{2}-5x-2=\left ( 3x+1 \right )\left ( x-2 \right )

      Check:   F\rightarrow 3x*x=3x^{2},\; O+I\rightarrow \left ( -6x \right )+1x=-5x,\; L\rightarrow \left ( +1 \right )\left ( -2 \right )=-2

                                b)\, \, 3x^{2}+5x-2=\left ( 3x-1 \right )\left ( x+2 \right )  

      Check:   F\rightarrow 3x*x=3x^{2},\; O+I\rightarrow \left ( 6x \right )+\left ( -1x \right )=+5x,\; L\rightarrow \left ( -1 \right )\left ( +2 \right )=-2 

6.  Factoring by grouping:  4 terms

2 & 2      Examples:    a)\, \, 2x^{3}-x^{2}-6x+3=x^{2}\left ( 2x-1 \right )-3\left ( 2x-1 \right )=\left ( x^{2}-3 \right )\left ( 2x-1 \right )

                                     b)\, \, 6x^{3}-2x+3x^{2}-1=2x\left ( 3x^{2}-1 \right )+( 3x^{2}-1 \right )=\left ( 2x+1 \right )\left ( 3x^{2} -1\right )

1&3        Examples:   a)\, \, x^{4}-2x-1-x^{2}=x^{4}-x^{2}-2x-1=x^{4}-\left ( x^{2}+2x+1 \right )=x^{4}-\left ( x+1 \right )^{2}=\left ( x^{2}+x+1 \right )\left ( x^{2}-x-1 \right )

                                    b)\, \, x^{4}+2x-1-x^{2}=x^{4}-x^{2}+2x-1=x^{4}-\left ( x^{2}-2x+1 \right )=x^{4}-\left ( x-1 \right )^{2}=\left ( x^{2}+x-1 \right )\left ( x^{2}-x+1 \right )

There are many more examples using factoring to simplify polynomial expressions.  The more you practice doing problems like these, the better you will get at doing them.  As your skills improve so will your ability to succeed in math. 

Filed under: Pre-requisites for Pre-Calculus

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