Question 644537
Here's an example:


Factor {{{5x^2 + 42x + 16}}}




Looking at the expression {{{5x^2+42x+16}}}, we can see that the first coefficient is {{{5}}}, the second coefficient is {{{42}}}, and the last term is {{{16}}}.



Now multiply the first coefficient {{{5}}} by the last term {{{16}}} to get {{{(5)(16)=80}}}.



Now the question is: what two whole numbers multiply to {{{80}}} (the previous product) <font size=4><b>and</b></font> add to the second coefficient {{{42}}}?



To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of {{{80}}} (the previous product).



Factors of {{{80}}}:

1,2,4,5,8,10,16,20,40,80

-1,-2,-4,-5,-8,-10,-16,-20,-40,-80



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



These factors pair up and multiply to {{{80}}}.

1*80 = 80
2*40 = 80
4*20 = 80
5*16 = 80
8*10 = 80
(-1)*(-80) = 80
(-2)*(-40) = 80
(-4)*(-20) = 80
(-5)*(-16) = 80
(-8)*(-10) = 80


Now let's add up each pair of factors to see if one pair adds to the middle coefficient {{{42}}}:



<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>80</font></td><td  align="center"><font color=black>1+80=81</font></td></tr><tr><td  align="center"><font color=red>2</font></td><td  align="center"><font color=red>40</font></td><td  align="center"><font color=red>2+40=42</font></td></tr><tr><td  align="center"><font color=black>4</font></td><td  align="center"><font color=black>20</font></td><td  align="center"><font color=black>4+20=24</font></td></tr><tr><td  align="center"><font color=black>5</font></td><td  align="center"><font color=black>16</font></td><td  align="center"><font color=black>5+16=21</font></td></tr><tr><td  align="center"><font color=black>8</font></td><td  align="center"><font color=black>10</font></td><td  align="center"><font color=black>8+10=18</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>-80</font></td><td  align="center"><font color=black>-1+(-80)=-81</font></td></tr><tr><td  align="center"><font color=black>-2</font></td><td  align="center"><font color=black>-40</font></td><td  align="center"><font color=black>-2+(-40)=-42</font></td></tr><tr><td  align="center"><font color=black>-4</font></td><td  align="center"><font color=black>-20</font></td><td  align="center"><font color=black>-4+(-20)=-24</font></td></tr><tr><td  align="center"><font color=black>-5</font></td><td  align="center"><font color=black>-16</font></td><td  align="center"><font color=black>-5+(-16)=-21</font></td></tr><tr><td  align="center"><font color=black>-8</font></td><td  align="center"><font color=black>-10</font></td><td  align="center"><font color=black>-8+(-10)=-18</font></td></tr></table>



From the table, we can see that the two numbers {{{2}}} and {{{40}}} add to {{{42}}} (the middle coefficient).



So the two numbers {{{2}}} and {{{40}}} both multiply to {{{80}}} <font size=4><b>and</b></font> add to {{{42}}}



Now replace the middle term {{{42x}}} with {{{2x+40x}}}. Remember, {{{2}}} and {{{40}}} add to {{{42}}}. So this shows us that {{{2x+40x=42x}}}.



{{{5x^2+highlight(2x+40x)+16}}} Replace the second term {{{42x}}} with {{{2x+40x}}}.



{{{(5x^2+2x)+(40x+16)}}} Group the terms into two pairs.



{{{x(5x+2)+(40x+16)}}} Factor out the GCF {{{x}}} from the first group.



{{{x(5x+2)+8(5x+2)}}} Factor out {{{8}}} 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+8)(5x+2)}}} Combine like terms. Or factor out the common term {{{5x+2}}}



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Answer:



So {{{5x^2+42x+16}}} factors to {{{(x+8)(5x+2)}}}.



In other words, {{{5x^2+42x+16=(x+8)(5x+2)}}}.



Note: you can check the answer by expanding {{{(x+8)(5x+2)}}} to get {{{5x^2+42x+16}}} or by graphing the original expression and the answer (the two graphs should be identical).


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