Question 635185


{{{y^3-14y^2+45y}}} Start with the given expression.



{{{y(y^2-14y+45)}}} Factor out the GCF {{{y}}}.



Now let's try to factor the inner expression {{{y^2-14y+45}}}



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Looking at the expression {{{y^2-14y+45}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{-14}}}, and the last term is {{{45}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{45}}} to get {{{(1)(45)=45}}}.



Now the question is: what two whole numbers multiply to {{{45}}} (the previous product) <font size=4><b>and</b></font> add to the second coefficient {{{-14}}}?



To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of {{{45}}} (the previous product).



Factors of {{{45}}}:

1,3,5,9,15,45

-1,-3,-5,-9,-15,-45



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



These factors pair up and multiply to {{{45}}}.

1*45 = 45
3*15 = 45
5*9 = 45
(-1)*(-45) = 45
(-3)*(-15) = 45
(-5)*(-9) = 45


Now let's add up each pair of factors to see if one pair adds to the middle coefficient {{{-14}}}:



<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>45</font></td><td  align="center"><font color=black>1+45=46</font></td></tr><tr><td  align="center"><font color=black>3</font></td><td  align="center"><font color=black>15</font></td><td  align="center"><font color=black>3+15=18</font></td></tr><tr><td  align="center"><font color=black>5</font></td><td  align="center"><font color=black>9</font></td><td  align="center"><font color=black>5+9=14</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>-45</font></td><td  align="center"><font color=black>-1+(-45)=-46</font></td></tr><tr><td  align="center"><font color=black>-3</font></td><td  align="center"><font color=black>-15</font></td><td  align="center"><font color=black>-3+(-15)=-18</font></td></tr><tr><td  align="center"><font color=red>-5</font></td><td  align="center"><font color=red>-9</font></td><td  align="center"><font color=red>-5+(-9)=-14</font></td></tr></table>



From the table, we can see that the two numbers {{{-5}}} and {{{-9}}} add to {{{-14}}} (the middle coefficient).



So the two numbers {{{-5}}} and {{{-9}}} both multiply to {{{45}}} <font size=4><b>and</b></font> add to {{{-14}}}



Now replace the middle term {{{-14y}}} with {{{-5y-9y}}}. Remember, {{{-5}}} and {{{-9}}} add to {{{-14}}}. So this shows us that {{{-5y-9y=-14y}}}.



{{{y^2+highlight(-5y-9y)+45}}} Replace the second term {{{-14y}}} with {{{-5y-9y}}}.



{{{(y^2-5y)+(-9y+45)}}} Group the terms into two pairs.



{{{y(y-5)+(-9y+45)}}} Factor out the GCF {{{y}}} from the first group.



{{{y(y-5)-9(y-5)}}} Factor out {{{9}}} 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-9)(y-5)}}} Combine like terms. Or factor out the common term {{{y-5}}}



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So {{{y(y^2-14y+45)}}} then factors further to {{{y(y-9)(y-5)}}}



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Answer:



So {{{y^3-14y^2+45y}}} completely factors to {{{y(y-9)(y-5)}}}.



In other words, {{{y^3-14y^2+45y=y(y-9)(y-5)}}}.



Note: you can check the answer by expanding {{{y(y-9)(y-5)}}} to get {{{y^3-14y^2+45y}}} or by graphing the original expression and the answer (the two graphs should be identical).