Question 610860


Looking at the expression {{{11w^2-14w+3}}}, we can see that the first coefficient is {{{11}}}, the second coefficient is {{{-14}}}, and the last term is {{{3}}}.



Now multiply the first coefficient {{{11}}} by the last term {{{3}}} to get {{{(11)(3)=33}}}.



Now the question is: what two whole numbers multiply to {{{33}}} (the previous product) <font size=4><b>and</b></font> add to the second coefficient {{{-14}}}?



To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of {{{33}}} (the previous product).



Factors of {{{33}}}:

1,3,11,33

-1,-3,-11,-33



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



These factors pair up and multiply to {{{33}}}.

1*33 = 33
3*11 = 33
(-1)*(-33) = 33
(-3)*(-11) = 33


Now let's add up each pair of factors to see if one pair adds to the middle coefficient {{{-14}}}:



<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>33</font></td><td  align="center"><font color=black>1+33=34</font></td></tr><tr><td  align="center"><font color=black>3</font></td><td  align="center"><font color=black>11</font></td><td  align="center"><font color=black>3+11=14</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>-33</font></td><td  align="center"><font color=black>-1+(-33)=-34</font></td></tr><tr><td  align="center"><font color=red>-3</font></td><td  align="center"><font color=red>-11</font></td><td  align="center"><font color=red>-3+(-11)=-14</font></td></tr></table>



From the table, we can see that the two numbers {{{-3}}} and {{{-11}}} add to {{{-14}}} (the middle coefficient).



So the two numbers {{{-3}}} and {{{-11}}} both multiply to {{{33}}} <font size=4><b>and</b></font> add to {{{-14}}}



Now replace the middle term {{{-14w}}} with {{{-3w-11w}}}. Remember, {{{-3}}} and {{{-11}}} add to {{{-14}}}. So this shows us that {{{-3w-11w=-14w}}}.



{{{11w^2+highlight(-3w-11w)+3}}} Replace the second term {{{-14w}}} with {{{-3w-11w}}}.



{{{(11w^2-3w)+(-11w+3)}}} Group the terms into two pairs.



{{{w(11w-3)+(-11w+3)}}} Factor out the GCF {{{w}}} from the first group.



{{{w(11w-3)-1(11w-3)}}} Factor out {{{1}}} 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.



{{{(w-1)(11w-3)}}} Combine like terms. Or factor out the common term {{{11w-3}}}



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Answer:



So {{{11w^2-14w+3}}} factors to {{{(w-1)(11w-3)}}}.



In other words, {{{11w^2-14w+3=(w-1)(11w-3)}}}.



Note: you can check the answer by expanding {{{(w-1)(11w-3)}}} to get {{{11w^2-14w+3}}} or by graphing the original expression and the answer (the two graphs should be identical).


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