<PRE><B>Question 884723</B>
Problem #1: Factoring {{{x^3+7x^2-x-7}}}



{{{x^3+7x^2-x-7}}} Start with the given expression.



{{{(x^3+7x^2)+(-x-7)}}} Group the terms



{{{x^2(x+7)+(-x-7)}}} Factor out {{{x^2}}} from the first group



{{{x^2(x+7)-1(x+7)}}} Factor out {{{-1}}} from the second group



{{{(x^2-1)(x+7)}}} Factor out {{{x+7}}}



{{{(x-1)(x+1)(x+7)}}} Factor {{{x^2-1}}} to get {{{(x-1)(x+1)}}} (difference of squares rule)



So {{{x^3+7x^2-x-7}}} factors to {{{(x-1)(x+1)(x+7)}}}


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Problem # 2: Factoring {{{3x^2+11x+10}}}



Looking at the expression {{{3x^2+11x+10}}}, we can see that the first coefficient is {{{3}}}, the second coefficient is {{{11}}}, and the last term is {{{10}}}.



Now multiply the first coefficient {{{3}}} by the last term {{{10}}} to get {{{(3)(10)=30}}}.



Now the question is: what two whole numbers multiply to {{{30}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{11}}}?



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 {{{30}}} (the previous product).



Factors of {{{30}}}:

1,2,3,5,6,10,15,30

-1,-2,-3,-5,-6,-10,-15,-30



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{30}}}.

1*30 = 30
2*15 = 30
3*10 = 30
5*6 = 30
(-1)*(-30) = 30
(-2)*(-15) = 30
(-3)*(-10) = 30
(-5)*(-6) = 30


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{11}}}:



&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;30&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+30=31&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;2&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;15&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;2+15=17&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;10&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;3+10=13&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;6&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;5+6=11&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;-30&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+(-30)=-31&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-2&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-15&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-2+(-15)=-17&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;-10&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-3+(-10)=-13&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;-6&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-5+(-6)=-11&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{5}}} and {{{6}}} add to {{{11}}} (the middle coefficient).



So the two numbers {{{5}}} and {{{6}}} both multiply to {{{30}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{11}}}



Now replace the middle term {{{11x}}} with {{{5x+6x}}}. Remember, {{{5}}} and {{{6}}} add to {{{11}}}. So this shows us that {{{5x+6x=11x}}}.



{{{3x^2+highlight(5x+6x)+10}}} Replace the second term {{{11x}}} with {{{5x+6x}}}.



{{{(3x^2+5x)+(6x+10)}}} Group the terms into two pairs.



{{{x(3x+5)+(6x+10)}}} Factor out the GCF {{{x}}} from the first group.



{{{x(3x+5)+2(3x+5)}}} Factor out {{{2}}} 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+2)(3x+5)}}} Combine like terms. Or factor out the common term {{{3x+5}}}



So {{{3x^2+11x+10}}} factors to {{{(x+2)(3x+5)}}}.



In other words, {{{3x^2+11x+10=(x+2)(3x+5)}}}.



Note: you can check the answer by expanding {{{(x+2)(3x+5)}}} to get {{{3x^2+11x+10}}} or by graphing the original expression and the answer (the two graphs should be identical).
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