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^6-3x^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^{2}&=&2x^{6} \\ &&\phantom{xxx}\blue{\text{the original equation}}\\
x^{2}-2x^{6}&=& 0 \\ &&\phantom{xxx}\blue{\text{reduced to }0}\\
x^{2}\cdot\left(1-2x^{4}\right)&=&0\\ &&\phantom{xxx}\blue{\text{factorized }}\\
x^{2}=0 &\lor& 1-2x^{4}=0\\ &&\phantom{xxx}\blue{A \cdot B=0 \Leftrightarrow A=0 \lor B=0}\\
x^{2}=0 &\lor& -2x^{4}=-1 \\ &&\phantom{xxx}\blue{\text{constants to the right hand side}}\\
x^{2}=0 &\lor& x^{4}=\frac{1}{2} \\ &&\phantom{xxx}\blue{\text{divided by the coefficient in front of term with }x}\\
x=0 &\lor& x=\sqrt[4]{\frac{1}{2} }\lor x=-\sqrt[4]{\frac{1}{2} }\\ &&\phantom{xxx}\blue{\text{root extracted on both sides}}\\
x=0 &\lor& x=\frac{1}{2}\sqrt[4]{8}\lor x=-\frac{1}{2}\sqrt[4]{8}\\ &&\phantom{xxx}\blue{\text{simplified}}
\end{array}#
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