Question 652174


{{{y^4+9y^3+8y^2}}} Start with the given expression.



{{{y^2(y^2+9y+8)}}} Factor out the GCF {{{y^2}}}.



Now let's try to factor the inner expression {{{y^2+9y+8}}}



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Looking at the expression {{{y^2+9y+8}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{9}}}, and the last term is {{{8}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{8}}} to get {{{(1)(8)=8}}}.



Now the question is: what two whole numbers multiply to {{{8}}} (the previous product) <font size=4><b>and</b></font> add to the second coefficient {{{9}}}?



To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of {{{8}}} (the previous product).



Factors of {{{8}}}:

1,2,4,8

-1,-2,-4,-8



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



These factors pair up and multiply to {{{8}}}.

1*8 = 8
2*4 = 8
(-1)*(-8) = 8
(-2)*(-4) = 8


Now let's add up each pair of factors to see if one pair adds to the middle coefficient {{{9}}}:



<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td  align="center"><font color=red>1</font></td><td  align="center"><font color=red>8</font></td><td  align="center"><font color=red>1+8=9</font></td></tr><tr><td  align="center"><font color=black>2</font></td><td  align="center"><font color=black>4</font></td><td  align="center"><font color=black>2+4=6</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>-8</font></td><td  align="center"><font color=black>-1+(-8)=-9</font></td></tr><tr><td  align="center"><font color=black>-2</font></td><td  align="center"><font color=black>-4</font></td><td  align="center"><font color=black>-2+(-4)=-6</font></td></tr></table>



From the table, we can see that the two numbers {{{1}}} and {{{8}}} add to {{{9}}} (the middle coefficient).



So the two numbers {{{1}}} and {{{8}}} both multiply to {{{8}}} <font size=4><b>and</b></font> add to {{{9}}}



Now replace the middle term {{{9y}}} with {{{y+8y}}}. Remember, {{{1}}} and {{{8}}} add to {{{9}}}. So this shows us that {{{y+8y=9y}}}.



{{{y^2+highlight(y+8y)+8}}} Replace the second term {{{9y}}} with {{{y+8y}}}.



{{{(y^2+y)+(8y+8)}}} Group the terms into two pairs.



{{{y(y+1)+(8y+8)}}} Factor out the GCF {{{y}}} from the first group.



{{{y(y+1)+8(y+1)}}} 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.



{{{(y+8)(y+1)}}} Combine like terms. Or factor out the common term {{{y+1}}}



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So {{{y^2(y^2+9y+8)}}} then factors further to {{{y^2(y+8)(y+1)}}}



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Answer:



So {{{y^4+9y^3+8y^2}}} completely factors to {{{y^2(y+8)(y+1)}}}.



In other words, {{{y^4+9y^3+8y^2=y^2(y+8)(y+1)}}}.



Note: you can check the answer by expanding {{{y^2(y+8)(y+1)}}} to get {{{y^4+9y^3+8y^2}}} or by graphing the original expression and the answer (the two graphs should be identical).


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