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Question
l^{3}-l^{4}-l^{5}
Factor the expression
l^{3}\left(1-l-l^{2}\right)
Evaluate
l^{3}-l^{4}-l^{5}
Rewrite the expression
l^{3}-l^{3}\times l-l^{3}\times l^{2}
Solution
l^{3}\left(1-l-l^{2}\right)
Show Solutions
Find the roots
l_{1}=-\frac{1+\sqrt{5}}{2},l_{2}=0,l_{3}=\frac{-1+\sqrt{5}}{2}
Alternative Form
l_{1}\approx -1.618034,l_{2}=0,l_{3}\approx 0.618034
Evaluate
l^{3}-l^{4}-l^{5}
To find the roots of the expression,set the expression equal to 0
l^{3}-l^{4}-l^{5}=0
Factor the expression
l^{3}\left(1-l-l^{2}\right)=0
\text{Separate the equation into }2\text{ possible cases}
\begin{align}&l^{3}=0\\&1-l-l^{2}=0\end{align}
The only way a power can be 0 is when the base equals 0
\begin{align}&l=0\\&1-l-l^{2}=0\end{align}
Solve the equation
More Steps
Evaluate
1-l-l^{2}=0
Rewrite in standard form
-l^{2}-l+1=0
Multiply both sides
l^{2}+l-1=0
\text{Substitute a=}1\text{,b=}1\text{ and c=}-1\text{ into the quadratic formula }l\text{=}\frac{-b\pm\sqrt{b^2-4ac}}{2a}
l=\frac{-1\pm \sqrt{1^{2}-4\left(-1\right)}}{2}
Simplify the expression
More Steps
Evaluate
1^{2}-4\left(-1\right)
1 raised to any power equals to 1
1-4\left(-1\right)
Simplify
1-\left(-4\right)
If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses
1+4
Add the numbers
5
l=\frac{-1\pm \sqrt{5}}{2}
\text{Separate the equation into }2\text{ possible cases}
\begin{align}&l=\frac{-1+\sqrt{5}}{2}\\&l=\frac{-1-\sqrt{5}}{2}\end{align}
\text{Use }\frac{-a}{b}=\frac{a}{-b}=-\frac{a}{b}\text{ to rewrite the fraction}
\begin{align}&l=\frac{-1+\sqrt{5}}{2}\\&l=-\frac{1+\sqrt{5}}{2}\end{align}
\begin{align}&l=0\\&l=\frac{-1+\sqrt{5}}{2}\\&l=-\frac{1+\sqrt{5}}{2}\end{align}
Solution
l_{1}=-\frac{1+\sqrt{5}}{2},l_{2}=0,l_{3}=\frac{-1+\sqrt{5}}{2}
Alternative Form
l_{1}\approx -1.618034,l_{2}=0,l_{3}\approx 0.618034
Show Solutions
