Question 151213

Let's say that we want to factor {{{4x^2-14x-30}}}



{{{4x^2-14x-30}}} Start with the given expression.



{{{2(2x^2-7x-15)}}} Factor out the GCF {{{2}}}




Now let's focus on the inner expression {{{2x^2-7x-15}}}



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



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



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



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



{{{x(2x+3)+(-10x-15)}}} Factor out the GCF {{{x}}} from the first group.



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



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



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



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Answer:



So {{{4x^2-14x-30}}} factors to {{{2(x-5)(2x+3)}}}.