Question 199751
Here's one way you can factor {{{ 2x^2 -13x +21}}}





Looking at the expression {{{2x^2-13x+21}}}, we can see that the first coefficient is {{{2}}}, the second coefficient is {{{-13}}}, and the last term is {{{21}}}.



Now multiply the first coefficient {{{2}}} by the last term {{{21}}} to get {{{(2)(21)=42}}}.



Now the question is: what two whole numbers multiply to {{{42}}} (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 {{{42}}} (the previous product).



Factors of {{{42}}}:

1,2,3,6,7,14,21,42

-1,-2,-3,-6,-7,-14,-21,-42



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



These factors pair up and multiply to {{{42}}}.

1*42
2*21
3*14
6*7
(-1)*(-42)
(-2)*(-21)
(-3)*(-14)
(-6)*(-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>42</font></td><td  align="center"><font color=black>1+42=43</font></td></tr><tr><td  align="center"><font color=black>2</font></td><td  align="center"><font color=black>21</font></td><td  align="center"><font color=black>2+21=23</font></td></tr><tr><td  align="center"><font color=black>3</font></td><td  align="center"><font color=black>14</font></td><td  align="center"><font color=black>3+14=17</font></td></tr><tr><td  align="center"><font color=black>6</font></td><td  align="center"><font color=black>7</font></td><td  align="center"><font color=black>6+7=13</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>-42</font></td><td  align="center"><font color=black>-1+(-42)=-43</font></td></tr><tr><td  align="center"><font color=black>-2</font></td><td  align="center"><font color=black>-21</font></td><td  align="center"><font color=black>-2+(-21)=-23</font></td></tr><tr><td  align="center"><font color=black>-3</font></td><td  align="center"><font color=black>-14</font></td><td  align="center"><font color=black>-3+(-14)=-17</font></td></tr><tr><td  align="center"><font color=red>-6</font></td><td  align="center"><font color=red>-7</font></td><td  align="center"><font color=red>-6+(-7)=-13</font></td></tr></table>



From the table, we can see that the two numbers {{{-6}}} and {{{-7}}} add to {{{-13}}} (the middle coefficient).



So the two numbers {{{-6}}} and {{{-7}}} both multiply to {{{42}}} <font size=4><b>and</b></font> add to {{{-13}}}



Now replace the middle term {{{-13x}}} with {{{-6x-7x}}}. Remember, {{{-6}}} and {{{-7}}} add to {{{-13}}}. So this shows us that {{{-6x-7x=-13x}}}.



{{{2x^2+highlight(-6x-7x)+21}}} Replace the second term {{{-13x}}} with {{{-6x-7x}}}.



{{{(2x^2-6x)+(-7x+21)}}} Group the terms into two pairs.



{{{2x(x-3)+(-7x+21)}}} Factor out the GCF {{{2x}}} from the first group.



{{{2x(x-3)-7(x-3)}}} 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.



{{{(2x-7)(x-3)}}} Combine like terms. Or factor out the common term {{{x-3}}}


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Answer:



So {{{2x^2-13x+21}}} factors to {{{(2x-7)(x-3)}}}.



Note: you can check the answer by FOILing {{{(2x-7)(x-3)}}} to get {{{2x^2-13x+21}}} or by graphing the original expression and the answer (the two graphs should be identical).





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