Question 592904


Looking at the expression {{{x^2-8x+15}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{-8}}}, and the last term is {{{15}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{15}}} to get {{{(1)(15)=15}}}.



Now the question is: what two whole numbers multiply to {{{15}}} (the previous product) <font size=4><b>and</b></font> add to the second coefficient {{{-8}}}?



To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of {{{15}}} (the previous product).



Factors of {{{15}}}:

1,3,5,15

-1,-3,-5,-15



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



These factors pair up and multiply to {{{15}}}.

1*15 = 15
3*5 = 15
(-1)*(-15) = 15
(-3)*(-5) = 15


Now let's add up each pair of factors to see if one pair adds to the middle coefficient {{{-8}}}:



<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>15</font></td><td  align="center"><font color=black>1+15=16</font></td></tr><tr><td  align="center"><font color=black>3</font></td><td  align="center"><font color=black>5</font></td><td  align="center"><font color=black>3+5=8</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>-15</font></td><td  align="center"><font color=black>-1+(-15)=-16</font></td></tr><tr><td  align="center"><font color=red>-3</font></td><td  align="center"><font color=red>-5</font></td><td  align="center"><font color=red>-3+(-5)=-8</font></td></tr></table>



From the table, we can see that the two numbers {{{-3}}} and {{{-5}}} add to {{{-8}}} (the middle coefficient).



So the two numbers {{{-3}}} and {{{-5}}} both multiply to {{{15}}} <font size=4><b>and</b></font> add to {{{-8}}}



Now replace the middle term {{{-8x}}} with {{{-3x-5x}}}. Remember, {{{-3}}} and {{{-5}}} add to {{{-8}}}. So this shows us that {{{-3x-5x=-8x}}}.



{{{x^2+highlight(-3x-5x)+15}}} Replace the second term {{{-8x}}} with {{{-3x-5x}}}.



{{{(x^2-3x)+(-5x+15)}}} Group the terms into two pairs.



{{{x(x-3)+(-5x+15)}}} Factor out the GCF {{{x}}} from the first group.



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



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Answer:



So {{{x(x^2-8x+15)}}} factors to {{{x(x-5)(x-3)}}}.



In other words, {{{x(x^2-8x+15)=x(x-5)(x-3)}}}.



Note: you can check the answer by expanding {{{x(x-5)(x-3)}}} to get {{{x(x^2-8x+15)}}} or by graphing the original expression and the answer (the two graphs should be identical).


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