Factor \({y}^{5}+243\) using Polynomial Division.
Factor the following.
\[{y}^{5}+243\]
First, find all factors of the constant term 243.
\[1, 3, 9, 27, 81, 243\]
Try each factor above using the Remainder Theorem.
Substitute 1 into y. Since the result is not 0, y-1 is not a factor..\({1}^{5}+243 = 244\)
Substitute -1 into y. Since the result is not 0, y+1 is not a factor..\({(-1)}^{5}+243 = 242\)
Substitute 3 into y. Since the result is not 0, y-3 is not a factor..\({3}^{5}+243 = 486\)
Substitute -3 into y. Since the result is 0, y+3 is a factor..\({(-3)}^{5}+243 = 0\)
\[y+3\]
Polynomial Division: Divide \({y}^{5}+243\) by \(y+3\).
| \[y^4\] | \[-3y^3\] | \[9y^2\] | \[-27y\] | \[81\] | |
| \[y+3\] | \[y^5\] | \[\] | \[\] | \[\] | \[\] | \[243\] |
| \[y^5\] | \[3y^4\] | | | | |
| | \[-3y^4\] | \[\] | \[\] | \[\] | \[243\] |
| | \[-3y^4\] | \[-9y^3\] | | | |
| | | \[9y^3\] | \[\] | \[\] | \[243\] |
| | | \[9y^3\] | \[27y^2\] | | |
| | | | \[-27y^2\] | \[\] | \[243\] |
| | | | \[-27y^2\] | \[-81y\] | |
| | | | | \[81y\] | \[243\] |
| | | | | \[81y\] | \[243\] |
| | | | | | \[\] |
Rewrite the expression using the above.
\[{y}^{4}-3{y}^{3}+9{y}^{2}-27y+81\]
l*IM*m*y->-3\(\frac{({y}^{4}-3{y}^{3}+9{y}^{2}-27y+81)(y+3)}{{y}^{3}+27}\)
Rewrite \({y}^{3}+27\) in the form \({a}^{3}+{b}^{3}\), where \(a=y\) and \(b=3\).
l*IM*m*y->-3\(\frac{({y}^{4}-3{y}^{3}+9{y}^{2}-27y+81)(y+3)}{{y}^{3}+{3}^{3}}\)
Use Sum of Cubes: \({a}^{3}+{b}^{3}=(a+b)({a}^{2}-ab+{b}^{2})\).
l*IM*m*y->-3\(\frac{({y}^{4}-3{y}^{3}+9{y}^{2}-27y+81)(y+3)}{(y+3)({y}^{2}-(y)(3)+{3}^{2})}\)
Simplify \({3}^{2}\) to \(9\).
l*IM*m*y->-3\(\frac{({y}^{4}-3{y}^{3}+9{y}^{2}-27y+81)(y+3)}{(y+3)({y}^{2}-y\times 3+9)}\)
Regroup terms.
l*IM*m*y->-3\(\frac{({y}^{4}-3{y}^{3}+9{y}^{2}-27y+81)(y+3)}{(y+3)({y}^{2}-3y+9)}\)
Regroup terms.
-+l*IM*m*y>-3\(\frac{({y}^{4}-3{y}^{3}+9{y}^{2}-27y+81)(y+3)}{(y+3)({y}^{2}-3y+9)}\)
Divide both sides by \(\imath \).
-+l*m*y>-(3\(\frac{({y}^{4}-3{y}^{3}+9{y}^{2}-27y+81)(y+3)}{(y+3)({y}^{2}-3y+9)}\))/IM
Simplify )/IM] to \(3\(\frac{({y}^{4}-3{y}^{3}+9{y}^{2}-27y+81)(y+3)}{\}\).
-+l*m*y>-3\((({y}^{4}-3{y}^{3}+9{y}^{2}-27y+81)(y+3))/\))
Divide both sides by \(m\).
-+l*y>-(3\((({y}^{4}-3{y}^{3}+9{y}^{2}-27y+81)(y+3))/\)))/m
Simplify ))/m] to \(3\(\frac{({y}^{4}-3{y}^{3}+9{y}^{2}-27y+81)(y+3)}{\}\).
-+l*y>-3\((({y}^{4}-3{y}^{3}+9{y}^{2}-27y+81)(y+3))/\))
Divide both sides by \(y\).
-+l>-(3\((({y}^{4}-3{y}^{3}+9{y}^{2}-27y+81)(y+3))/\)))/y
Simplify ))/y] to \(3\(\frac{({y}^{4}-3{y}^{3}+9{y}^{2}-27y+81)(y+3)}{\}\).
-+l>-3\((({y}^{4}-3{y}^{3}+9{y}^{2}-27y+81)(y+3))/\))
Multiply both sides by \(-1\).
l<3\((({y}^{4}-3{y}^{3}+9{y}^{2}-27y+81)(y+3))/\))
l<3[((y^4-3*y^3+9*y^2-27*y+81)*(y+3))/(IM*m*y*(y+3)*(y^2-3*y+9)])
