Similar to solving quadratics we have a few methods to choose between when solving cubic equations:
- Solve using algebraic manipulation - For cubics such as $2x^3+16=0$2x3+16=0.
- Factorise - Fully factorising the cubic we can then use the null factor law to solve. If $a\times b\times c=0$a×b×c=0 then either $a=0$a=0 or $b=0$b=0 or $c=0$c=0.
- Technology - Once we have extracted the important information from a question and formed an equation, we could use our calculator to solve the equation graphically or algebraically.
When factorising a cubic recall we can factorise by:
- using the highest common factor
- factor special forms such as the sum and difference of cubes
- identifying a single factor, then using division to establish the remaining quadratic. From here you would employ any of the factorising methods for quadratics.
Practice questions
Question 1
Solve the equation $x^3=-8$x3=−8.
QUESTION 2
Solve the equation $\left(x+8\right)\left(x+4\right)\left(1+x\right)=0$(x+8)(x+4)(1+x)=0.
State the solutions on the same line, separated by commas.
QUESTION 3
Solve the equation $x^3-49x=0$x3−49x=0.
State the solutions on the same line, separated by a comma.
QUESTION 4
The cubic $P\left(x\right)=x^3-2x^2-5x+6$P(x)=x3−2x2−5x+6 has a factor of $x-3$x−3.
Solve for the roots of the cubic. If there is more than one root, state the solutions on the same line separated by commas.
QUESTION 5
Use technology to solve the cubic equation $48x^3-212x^2+84x+209=-36$48x3−212x2+84x+209=−36.
If there is more than one root, state the solutions on the same line separated by commas.