Question 704903


Looking at the expression {{{x^2-84x+83}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{-84}}}, and the last term is {{{83}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{83}}} to get {{{(1)(83)=83}}}.



Now the question is: what two whole numbers multiply to {{{83}}} (the previous product) <font size=4><b>and</b></font> add to the second coefficient {{{-84}}}?



To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of {{{83}}} (the previous product).



Factors of {{{83}}}:

1,83

-1,-83



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



These factors pair up and multiply to {{{83}}}.

1*83 = 83
(-1)*(-83) = 83


Now let's add up each pair of factors to see if one pair adds to the middle coefficient {{{-84}}}:



<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>83</font></td><td  align="center"><font color=black>1+83=84</font></td></tr><tr><td  align="center"><font color=red>-1</font></td><td  align="center"><font color=red>-83</font></td><td  align="center"><font color=red>-1+(-83)=-84</font></td></tr></table>



From the table, we can see that the two numbers {{{-1}}} and {{{-83}}} add to {{{-84}}} (the middle coefficient).



So the two numbers {{{-1}}} and {{{-83}}} both multiply to {{{83}}} <font size=4><b>and</b></font> add to {{{-84}}}



Now replace the middle term {{{-84x}}} with {{{-x-83x}}}. Remember, {{{-1}}} and {{{-83}}} add to {{{-84}}}. So this shows us that {{{-x-83x=-84x}}}.



{{{x^2+highlight(-x-83x)+83}}} Replace the second term {{{-84x}}} with {{{-x-83x}}}.



{{{(x^2-x)+(-83x+83)}}} Group the terms into two pairs.



{{{x(x-1)+(-83x+83)}}} Factor out the GCF {{{x}}} from the first group.



{{{x(x-1)-83(x-1)}}} Factor out {{{83}}} 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-83)(x-1)}}} Combine like terms. Or factor out the common term {{{x-1}}}



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Answer:



So {{{x^2-84x+83}}} factors to {{{(x-83)(x-1)}}}.



In other words, {{{x^2-84x+83=(x-83)(x-1)}}}.



Note: you can check the answer by expanding {{{(x-83)(x-1)}}} to get {{{x^2-84x+83}}} or by graphing the original expression and the answer (the two graphs should be identical).