Question 657729
Please post *one* problem at a time. Thanks.


I'll do #1 to get you started


# 1


Looking at the expression {{{x^2-16x+63}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{-16}}}, and the last term is {{{63}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{63}}} to get {{{(1)(63)=63}}}.



Now the question is: what two whole numbers multiply to {{{63}}} (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 {{{63}}} (the previous product).



Factors of {{{63}}}:

1,3,7,9,21,63

-1,-3,-7,-9,-21,-63



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



These factors pair up and multiply to {{{63}}}.

1*63 = 63
3*21 = 63
7*9 = 63
(-1)*(-63) = 63
(-3)*(-21) = 63
(-7)*(-9) = 63


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>63</font></td><td  align="center"><font color=black>1+63=64</font></td></tr><tr><td  align="center"><font color=black>3</font></td><td  align="center"><font color=black>21</font></td><td  align="center"><font color=black>3+21=24</font></td></tr><tr><td  align="center"><font color=black>7</font></td><td  align="center"><font color=black>9</font></td><td  align="center"><font color=black>7+9=16</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>-63</font></td><td  align="center"><font color=black>-1+(-63)=-64</font></td></tr><tr><td  align="center"><font color=black>-3</font></td><td  align="center"><font color=black>-21</font></td><td  align="center"><font color=black>-3+(-21)=-24</font></td></tr><tr><td  align="center"><font color=red>-7</font></td><td  align="center"><font color=red>-9</font></td><td  align="center"><font color=red>-7+(-9)=-16</font></td></tr></table>



From the table, we can see that the two numbers {{{-7}}} and {{{-9}}} add to {{{-16}}} (the middle coefficient).



So the two numbers {{{-7}}} and {{{-9}}} both multiply to {{{63}}} <font size=4><b>and</b></font> add to {{{-16}}}



Now replace the middle term {{{-16x}}} with {{{-7x-9x}}}. Remember, {{{-7}}} and {{{-9}}} add to {{{-16}}}. So this shows us that {{{-7x-9x=-16x}}}.



{{{x^2+highlight(-7x-9x)+63}}} Replace the second term {{{-16x}}} with {{{-7x-9x}}}.



{{{(x^2-7x)+(-9x+63)}}} Group the terms into two pairs.



{{{x(x-7)+(-9x+63)}}} Factor out the GCF {{{x}}} from the first group.



{{{x(x-7)-9(x-7)}}} Factor out {{{9}}} 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-9)(x-7)}}} Combine like terms. Or factor out the common term {{{x-7}}}



===============================================================



Answer:



So {{{x^2-16x+63}}} factors to {{{(x-9)(x-7)}}}.



In other words, {{{x^2-16x+63=(x-9)(x-7)}}}.



Note: you can check the answer by expanding {{{(x-9)(x-7)}}} to get {{{x^2-16x+63}}} or by graphing the original expression and the answer (the two graphs should be identical).


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