Question 355005


Looking at the expression {{{w^2-7w+12}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{-7}}}, and the last term is {{{12}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{12}}} to get {{{(1)(12)=12}}}.



Now the question is: what two whole numbers multiply to {{{12}}} (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 {{{12}}} (the previous product).



Factors of {{{12}}}:

1,2,3,4,6,12

-1,-2,-3,-4,-6,-12



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



These factors pair up and multiply to {{{12}}}.

1*12 = 12
2*6 = 12
3*4 = 12
(-1)*(-12) = 12
(-2)*(-6) = 12
(-3)*(-4) = 12


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>12</font></td><td  align="center"><font color=black>1+12=13</font></td></tr><tr><td  align="center"><font color=black>2</font></td><td  align="center"><font color=black>6</font></td><td  align="center"><font color=black>2+6=8</font></td></tr><tr><td  align="center"><font color=black>3</font></td><td  align="center"><font color=black>4</font></td><td  align="center"><font color=black>3+4=7</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>-12</font></td><td  align="center"><font color=black>-1+(-12)=-13</font></td></tr><tr><td  align="center"><font color=black>-2</font></td><td  align="center"><font color=black>-6</font></td><td  align="center"><font color=black>-2+(-6)=-8</font></td></tr><tr><td  align="center"><font color=red>-3</font></td><td  align="center"><font color=red>-4</font></td><td  align="center"><font color=red>-3+(-4)=-7</font></td></tr></table>



From the table, we can see that the two numbers {{{-3}}} and {{{-4}}} add to {{{-7}}} (the middle coefficient).



So the two numbers {{{-3}}} and {{{-4}}} both multiply to {{{12}}} <font size=4><b>and</b></font> add to {{{-7}}}



Now replace the middle term {{{-7w}}} with {{{-3w-4w}}}. Remember, {{{-3}}} and {{{-4}}} add to {{{-7}}}. So this shows us that {{{-3w-4w=-7w}}}.



{{{w^2+highlight(-3w-4w)+12}}} Replace the second term {{{-7w}}} with {{{-3w-4w}}}.



{{{(w^2-3w)+(-4w+12)}}} Group the terms into two pairs.



{{{w(w-3)+(-4w+12)}}} Factor out the GCF {{{w}}} from the first group.



{{{w(w-3)-4(w-3)}}} Factor out {{{4}}} 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-4)(w-3)}}} Combine like terms. Or factor out the common term {{{w-3}}}



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Answer:



So {{{w^2-7w+12}}} factors to {{{(w-4)(w-3)}}}.



In other words, {{{w^2-7w+12=(w-4)(w-3)}}}.



Note: you can check the answer by expanding {{{(w-4)(w-3)}}} to get {{{w^2-7w+12}}} or by graphing the original expression and the answer (the two graphs should be identical).



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Jim