<PRE><B>Question 725822</B>


Looking at the expression {{{3x^2+8x-3}}}, we can see that the first coefficient is {{{3}}}, the second coefficient is {{{8}}}, and the last term is {{{-3}}}.



Now multiply the first coefficient {{{3}}} by the last term {{{-3}}} to get {{{(3)(-3)=-9}}}.



Now the question is: what two whole numbers multiply to {{{-9}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{8}}}?



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 {{{-9}}} (the previous product).



Factors of {{{-9}}}:

1,3,9

-1,-3,-9



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{-9}}}.

1*(-9) = -9
3*(-3) = -9
(-1)*(9) = -9
(-3)*(3) = -9


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{8}}}:



&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;-9&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+(-9)=-8&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;-3&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;3+(-3)=0&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-1&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;9&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-1+9=8&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;3&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-3+3=0&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{-1}}} and {{{9}}} add to {{{8}}} (the middle coefficient).



So the two numbers {{{-1}}} and {{{9}}} both multiply to {{{-9}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{8}}}



Now replace the middle term {{{8x}}} with {{{-x+9x}}}. Remember, {{{-1}}} and {{{9}}} add to {{{8}}}. So this shows us that {{{-x+9x=8x}}}.



{{{3x^2+highlight(-x+9x)-3}}} Replace the second term {{{8x}}} with {{{-x+9x}}}.



{{{(3x^2-x)+(9x-3)}}} Group the terms into two pairs.



{{{x(3x-1)+(9x-3)}}} Factor out the GCF {{{x}}} from the first group.



{{{x(3x-1)+3(3x-1)}}} Factor out {{{3}}} 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+3)(3x-1)}}} Combine like terms. Or factor out the common term {{{3x-1}}}



===============================================================



Answer:



So {{{3x^2+8x-3}}} factors to {{{(x+3)(3x-1)}}}.



In other words, {{{3x^2+8x-3=(x+3)(3x-1)}}}.



Note: you can check the answer by expanding {{{(x+3)(3x-1)}}} to get {{{3x^2+8x-3}}} or by graphing the original expression and the answer (the two graphs should be identical).
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