Question 319179


{{{-x^2+16x+57}}} Start with the given expression.



{{{-(x^2-16x-57)}}} Factor out the GCF {{{-1}}}.



Now let's try to factor the inner expression {{{x^2-16x-57}}}



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Looking at the expression {{{x^2-16x-57}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{-16}}}, and the last term is {{{-57}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{-57}}} to get {{{(1)(-57)=-57}}}.



Now the question is: what two whole numbers multiply to {{{-57}}} (the previous product) <font size=4><b>and</b></font> add to the second coefficient {{{-16}}}?



To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of {{{-57}}} (the previous product).



Factors of {{{-57}}}:

1,3,19,57

-1,-3,-19,-57



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



These factors pair up and multiply to {{{-57}}}.

1*(-57) = -57
3*(-19) = -57
(-1)*(57) = -57
(-3)*(19) = -57


Now let's add up each pair of factors to see if one pair adds to the middle coefficient {{{-16}}}:



<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>-57</font></td><td  align="center"><font color=black>1+(-57)=-56</font></td></tr><tr><td  align="center"><font color=red>3</font></td><td  align="center"><font color=red>-19</font></td><td  align="center"><font color=red>3+(-19)=-16</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>57</font></td><td  align="center"><font color=black>-1+57=56</font></td></tr><tr><td  align="center"><font color=black>-3</font></td><td  align="center"><font color=black>19</font></td><td  align="center"><font color=black>-3+19=16</font></td></tr></table>



From the table, we can see that the two numbers {{{3}}} and {{{-19}}} add to {{{-16}}} (the middle coefficient).



So the two numbers {{{3}}} and {{{-19}}} both multiply to {{{-57}}} <font size=4><b>and</b></font> add to {{{-16}}}



Now replace the middle term {{{-16x}}} with {{{3x-19x}}}. Remember, {{{3}}} and {{{-19}}} add to {{{-16}}}. So this shows us that {{{3x-19x=-16x}}}.



{{{x^2+highlight(3x-19x)-57}}} Replace the second term {{{-16x}}} with {{{3x-19x}}}.



{{{(x^2+3x)+(-19x-57)}}} Group the terms into two pairs.



{{{x(x+3)+(-19x-57)}}} Factor out the GCF {{{x}}} from the first group.



{{{x(x+3)-19(x+3)}}} Factor out {{{19}}} 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-19)(x+3)}}} Combine like terms. Or factor out the common term {{{x+3}}}



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So {{{-1(x^2-16x-57)}}} then factors further to {{{-(x-19)(x+3)}}}



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Answer:



So {{{-x^2+16x+57}}} completely factors to {{{-(x-19)(x+3)}}}.



In other words, {{{-x^2+16x+57=-(x-19)(x+3)}}}.



Note: you can check the answer by expanding {{{-(x-19)(x+3)}}} to get {{{-x^2+16x+57}}} or by graphing the original expression and the answer (the two graphs should be identical).