Question 156342


Looking at the expression {{{15x^2+7x-2}}}, we can see that the first coefficient is {{{15}}}, the second coefficient is {{{7}}}, and the last term is {{{-2}}}.



Now multiply the first coefficient {{{15}}} by the last term {{{-2}}} to get {{{(15)(-2)=-30}}}.



Now the question is: what two whole numbers multiply to {{{-30}}} (the previous product) <font size=4><b>and</b></font> add to the second coefficient {{{7}}}?



To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of {{{-30}}} (the previous product).



Factors of {{{-30}}}:

1,2,3,5,6,10,15,30

-1,-2,-3,-5,-6,-10,-15,-30



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



These factors pair up and multiply to {{{-30}}}.

1*(-30)
2*(-15)
3*(-10)
5*(-6)
(-1)*(30)
(-2)*(15)
(-3)*(10)
(-5)*(6)


Now let's add up each pair of factors to see if one pair adds to the middle coefficient {{{7}}}:



<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>-30</font></td><td  align="center"><font color=black>1+(-30)=-29</font></td></tr><tr><td  align="center"><font color=black>2</font></td><td  align="center"><font color=black>-15</font></td><td  align="center"><font color=black>2+(-15)=-13</font></td></tr><tr><td  align="center"><font color=black>3</font></td><td  align="center"><font color=black>-10</font></td><td  align="center"><font color=black>3+(-10)=-7</font></td></tr><tr><td  align="center"><font color=black>5</font></td><td  align="center"><font color=black>-6</font></td><td  align="center"><font color=black>5+(-6)=-1</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>30</font></td><td  align="center"><font color=black>-1+30=29</font></td></tr><tr><td  align="center"><font color=black>-2</font></td><td  align="center"><font color=black>15</font></td><td  align="center"><font color=black>-2+15=13</font></td></tr><tr><td  align="center"><font color=red>-3</font></td><td  align="center"><font color=red>10</font></td><td  align="center"><font color=red>-3+10=7</font></td></tr><tr><td  align="center"><font color=black>-5</font></td><td  align="center"><font color=black>6</font></td><td  align="center"><font color=black>-5+6=1</font></td></tr></table>



From the table, we can see that the two numbers {{{-3}}} and {{{10}}} add to {{{7}}} (the middle coefficient).



So the two numbers {{{-3}}} and {{{10}}} both multiply to {{{-30}}} <font size=4><b>and</b></font> add to {{{7}}}



Now replace the middle term {{{7x}}} with {{{-3x+10x}}}. Remember, {{{-3}}} and {{{10}}} add to {{{7}}}. So this shows us that {{{-3x+10x=7x}}}.



{{{15x^2+highlight(-3x+10x)-2}}} Replace the second term {{{7x}}} with {{{-3x+10x}}}.



{{{(15x^2-3x)+(10x-2)}}} Group the terms into two pairs.



{{{3x(5x-1)+(10x-2)}}} Factor out the GCF {{{3x}}} from the first group.



{{{3x(5x-1)+2(5x-1)}}} Factor out {{{2}}} 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.



{{{(3x+2)(5x-1)}}} Combine like terms. Or factor out the common term {{{5x-1}}}


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Answer:



So {{{15x^2+7x-2}}} factors to {{{(3x+2)(5x-1)}}}.



Note: you can check the answer by FOILing {{{(3x+2)(5x-1)}}} to get {{{15x^2+7x-2}}} or by graphing the original expression and the answer (the two graphs should be identical).