<PRE><B>Question 191799</B>


Looking at the expression {{{x^2+22x+57}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{22}}}, 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) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{22}}}?



To find these two numbers, we need to list &lt;font size=4&gt;&lt;b&gt;all&lt;/b&gt;&lt;/font&gt; 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
3*19
(-1)*(-57)
(-3)*(-19)


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{22}}}:



&lt;table border=&quot;1&quot;&gt;&lt;th&gt;First Number&lt;/th&gt;&lt;th&gt;Second Number&lt;/th&gt;&lt;th&gt;Sum&lt;/th&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;57&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+57=58&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;3&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;19&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;3+19=22&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-57&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+(-57)=-58&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-3&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-19&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-3+(-19)=-22&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{3}}} and {{{19}}} add to {{{22}}} (the middle coefficient).



So the two numbers {{{3}}} and {{{19}}} both multiply to {{{57}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{22}}}



Now replace the middle term {{{22x}}} with {{{3x+19x}}}. Remember, {{{3}}} and {{{19}}} add to {{{22}}}. So this shows us that {{{3x+19x=22x}}}.



{{{x^2+highlight(3x+19x)+57}}} Replace the second term {{{22x}}} 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}}}


---------------------------------------------



Answer:



So {{{x^2+22x+57}}} factors to {{{(x+19)(x+3)}}}.



Note: you can check the answer by FOILing {{{(x+19)(x+3)}}} to get {{{x^2+22x+57}}} or by graphing the original expression and the answer (the two graphs should be identical).
</PRE>