Functions: Higher degree polynomials
Solving higher degree polynomials with factorization
Factoring out
Procedure We solve an equation with a polynomial by factoring out. 
Example #x^4+3x^3+2x^2=0# 

Step 1 
Factor out the biggest possible term. 
#x^2\left(x^2+3x+2\right)=0# 
Step 2 
Use the rule #A \cdot B=0# giving #A=0 \lor B=0#. 
#x^2=0 \lor x^2+3x+2=0# 
Step 3 
Solve the obtained equations. 
#x=0 \lor x=2 \lor x=1# 
Factorization
Procedure We solve an equation with polynomials in #x# by means of factorization. 
Example #x^63x^3+2=0# 

Step 1 
Write the equation as #a x^{\blue n \cdot 2}+b{x^\blue n}+c=0#. 
#x^{\blue3 \cdot 2}3{x^\blue3}+2=0# 
Step 2 
Now factorize the left hand side. 
#\left(x^{\blue3}2\right) \left( x^{\blue3}1\right) =0# 
Step 3 
Use the rule #A \cdot B=0# giving #A=0 \lor B=0#. 
#x^{\blue3}2=0 \lor x^{\blue3}1=0# 
Step 4 
Reduce both equations to the form #x^{\blue n}=c#, in which #c# is a number. 
#{x^\blue3}=2\lor {x^\blue3}=1# 
Stap 5 
Use higher degree roots to solve the obtained equations. 
#x=\sqrt[3]{2}\lor x=1# 
#\begin{array}{rcl}x^{3}&=&3x^{7} \\ &&\phantom{xxx}\blue{\text{the original equation}}\\
x^{3}3x^{7}&=& 0 \\ &&\phantom{xxx}\blue{\text{reduced to }0}\\
x^{3}\cdot\left(13x^{4}\right)&=&0\\ &&\phantom{xxx}\blue{\text{factorized }}\\
x^{3}=0 &\lor& 13x^{4}=0\\ &&\phantom{xxx}\blue{A \cdot B=0 \Leftrightarrow A=0 \lor B=0}\\
x^{3}=0 &\lor& 3x^{4}=1 \\ &&\phantom{xxx}\blue{\text{constants to the right hand side}}\\
x^{3}=0 &\lor& x^{4}=\frac{1}{3} \\ &&\phantom{xxx}\blue{\text{divided by the coefficient in front of term with }x}\\
x=0 &\lor& x=\sqrt[4]{\frac{1}{3} }\lor x=\sqrt[4]{\frac{1}{3} }\\ &&\phantom{xxx}\blue{\text{root extracted on both sides}}\\
x=0 &\lor& x=\frac{1}{3}\sqrt[4]{27}\lor x=\frac{1}{3}\sqrt[4]{27}\\ &&\phantom{xxx}\blue{\text{simplified}}
\end{array}#
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