<PRE><B>Question 191090</B>

Looking at the expression {{{5x^2-12x+7}}}, we can see that the first coefficient is {{{5}}}, the second coefficient is {{{-12}}}, and the last term is {{{7}}}.



Now multiply the first coefficient {{{5}}} by the last term {{{7}}} to get {{{(5)(7)=35}}}.



Now the question is: what two whole numbers multiply to {{{35}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{-12}}}?



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 {{{35}}} (the previous product).



Factors of {{{35}}}:

1,5,7,35

-1,-5,-7,-35



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{35}}}.

1*35
5*7
(-1)*(-35)
(-5)*(-7)


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{-12}}}:



&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;35&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+35=36&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;5&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;7&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;5+7=12&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;-35&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+(-35)=-36&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-5&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-7&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-5+(-7)=-12&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{-5}}} and {{{-7}}} add to {{{-12}}} (the middle coefficient).



So the two numbers {{{-5}}} and {{{-7}}} both multiply to {{{35}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{-12}}}



Now replace the middle term {{{-12x}}} with {{{-5x-7x}}}. Remember, {{{-5}}} and {{{-7}}} add to {{{-12}}}. So this shows us that {{{-5x-7x=-12x}}}.



{{{5x^2+highlight(-5x-7x)+7}}} Replace the second term {{{-12x}}} with {{{-5x-7x}}}.



{{{(5x^2-5x)+(-7x+7)}}} Group the terms into two pairs.



{{{5x(x-1)+(-7x+7)}}} Factor out the GCF {{{5x}}} from the first group.



{{{5x(x-1)-7(x-1)}}} Factor out {{{7}}} 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.



{{{(5x-7)(x-1)}}} Combine like terms. Or factor out the common term {{{x-1}}}


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Answer:



So {{{5x^2-12x+7}}} factors to {{{(5x-7)(x-1)}}}.



Note: you can check the answer by FOILing {{{(5x-7)(x-1)}}} to get {{{5x^2-12x+7}}} or by graphing the original expression and the answer (the two graphs should be identical).
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