Question 157254
Do you want to factor this?





Looking at the expression {{{2b^2+13b-7}}}, we can see that the first coefficient is {{{2}}}, the second coefficient is {{{13}}}, and the last term is {{{-7}}}.



Now multiply the first coefficient {{{2}}} by the last term {{{-7}}} to get {{{(2)(-7)=-14}}}.



Now the question is: what two whole numbers multiply to {{{-14}}} (the previous product) <font size=4><b>and</b></font> add to the second coefficient {{{13}}}?



To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of {{{-14}}} (the previous product).



Factors of {{{-14}}}:

1,2,7,14

-1,-2,-7,-14



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



These factors pair up and multiply to {{{-14}}}.

1*(-14)
2*(-7)
(-1)*(14)
(-2)*(7)


Now let's add up each pair of factors to see if one pair adds to the middle coefficient {{{13}}}:



<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>-14</font></td><td  align="center"><font color=black>1+(-14)=-13</font></td></tr><tr><td  align="center"><font color=black>2</font></td><td  align="center"><font color=black>-7</font></td><td  align="center"><font color=black>2+(-7)=-5</font></td></tr><tr><td  align="center"><font color=red>-1</font></td><td  align="center"><font color=red>14</font></td><td  align="center"><font color=red>-1+14=13</font></td></tr><tr><td  align="center"><font color=black>-2</font></td><td  align="center"><font color=black>7</font></td><td  align="center"><font color=black>-2+7=5</font></td></tr></table>



From the table, we can see that the two numbers {{{-1}}} and {{{14}}} add to {{{13}}} (the middle coefficient).



So the two numbers {{{-1}}} and {{{14}}} both multiply to {{{-14}}} <font size=4><b>and</b></font> add to {{{13}}}



Now replace the middle term {{{13b}}} with {{{-b+14b}}}. Remember, {{{-1}}} and {{{14}}} add to {{{13}}}. So this shows us that {{{-b+14b=13b}}}.



{{{2b^2+highlight(-b+14b)-7}}} Replace the second term {{{13b}}} with {{{-b+14b}}}.



{{{(2b^2-b)+(14b-7)}}} Group the terms into two pairs.



{{{b(2b-1)+(14b-7)}}} Factor out the GCF {{{b}}} from the first group.



{{{b(2b-1)+7(2b-1)}}} Factor out {{{7}}} 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.



{{{(b+7)(2b-1)}}} Combine like terms. Or factor out the common term {{{2b-1}}}


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Answer:



So {{{2b^2+13b-7}}} factors to {{{(b+7)(2b-1)}}}.



In other words, {{{2b^2+13b-7=(b+7)(2b-1)}}}



Note: you can check the answer by FOILing {{{(b+7)(2b-1)}}} to get {{{2b^2+13b-7}}} or by graphing the original expression and the answer (the two graphs should be identical).




Questions? Email me at <a href="mailto:jim_thompson5910@hotmail.com?subject=Algebra Help">jim_thompson5910@hotmail.com</a> (note: the space is really an underscore)