Question 200535

Looking at the expression {{{8x^2-10x-3}}}, we can see that the first coefficient is {{{8}}}, the second coefficient is {{{-10}}}, and the last term is {{{-3}}}.



Now multiply the first coefficient {{{8}}} by the last term {{{-3}}} to get {{{(8)(-3)=-24}}}.



Now the question is: what two whole numbers multiply to {{{-24}}} (the previous product) <font size=4><b>and</b></font> add to the second coefficient {{{-10}}}?



To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of {{{-24}}} (the previous product).



Factors of {{{-24}}}:

1,2,3,4,6,8,12,24

-1,-2,-3,-4,-6,-8,-12,-24



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{-24}}}.

1*(-24)
2*(-12)
3*(-8)
4*(-6)
(-1)*(24)
(-2)*(12)
(-3)*(8)
(-4)*(6)


Now let's add up each pair of factors to see if one pair adds to the middle coefficient {{{-10}}}:



<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td  align="center"><font color=black>1</font></td><td  align="center"><font color=black>-24</font></td><td  align="center"><font color=black>1+(-24)=-23</font></td></tr><tr><td  align="center"><font color=red>2</font></td><td  align="center"><font color=red>-12</font></td><td  align="center"><font color=red>2+(-12)=-10</font></td></tr><tr><td  align="center"><font color=black>3</font></td><td  align="center"><font color=black>-8</font></td><td  align="center"><font color=black>3+(-8)=-5</font></td></tr><tr><td  align="center"><font color=black>4</font></td><td  align="center"><font color=black>-6</font></td><td  align="center"><font color=black>4+(-6)=-2</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>24</font></td><td  align="center"><font color=black>-1+24=23</font></td></tr><tr><td  align="center"><font color=black>-2</font></td><td  align="center"><font color=black>12</font></td><td  align="center"><font color=black>-2+12=10</font></td></tr><tr><td  align="center"><font color=black>-3</font></td><td  align="center"><font color=black>8</font></td><td  align="center"><font color=black>-3+8=5</font></td></tr><tr><td  align="center"><font color=black>-4</font></td><td  align="center"><font color=black>6</font></td><td  align="center"><font color=black>-4+6=2</font></td></tr></table>



From the table, we can see that the two numbers {{{2}}} and {{{-12}}} add to {{{-10}}} (the middle coefficient).



So the two numbers {{{2}}} and {{{-12}}} both multiply to {{{-24}}} <font size=4><b>and</b></font> add to {{{-10}}}



Now replace the middle term {{{-10x}}} with {{{2x-12x}}}. Remember, {{{2}}} and {{{-12}}} add to {{{-10}}}. So this shows us that {{{2x-12x=-10x}}}.



{{{8x^2+highlight(2x-12x)-3}}} Replace the second term {{{-10x}}} with {{{2x-12x}}}.



{{{(8x^2+2x)+(-12x-3)}}} Group the terms into two pairs.



{{{2x(4x+1)+(-12x-3)}}} Factor out the GCF {{{2x}}} from the first group.



{{{2x(4x+1)-3(4x+1)}}} Factor out {{{3}}} from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.



{{{(2x-3)(4x+1)}}} Combine like terms. Or factor out the common term {{{4x+1}}}


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Answer:



So {{{8x^2-10x-3}}} factors to {{{(2x-3)(4x+1)}}}.



Note: you can check the answer by FOILing {{{(2x-3)(4x+1)}}} to get {{{8x^2-10x-3}}} or by graphing the original expression and the answer (the two graphs should be identical).



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