We have already seen how to solve polynomials up to cubics in Solve cubic equations, using skills we learnt in Factorising cubic equations. We can apply some of these skills to solving polynomials of degree four by using the following methods.
- Solve using algebraic manipulation - For quartics such as $2\left(x-16\right)^4=0$2(x−16)4=0.
- Factorise - Fully factorising the polynomial and then we can then use the factor theorem to solve. If $a\times b=0$a×b=0 then either $a=0$a=0 or $b=0$b=0.
- Technology - Once we have extracted the important information from a question and formed an equation, we could use technology to solve the equation graphically or algebraically.
Worked examples
Example 1
The polynomial $x^4-6x^2+8$x4−6x2+8 has roots at $x=2$x=2 and $x=-2$x=−2 satisfying $P(x)=0$P(x)=0.
Solve for the other roots of $P(x)=0$P(x)=0.
Think: By the factor theorem, since $2$2 and $-2$−2 are roots of $P(x)=0$P(x)=0, it follows that $x-2$x−2 and $x+2$x+2 are factors of $P(x)$P(x). Their product $\left(x-2\right)\left(x+2\right)=x^2-4$(x−2)(x+2)=x2−4 is also a factor.
Do:
| $x^4-6x^2+8$x4−6x2+8 | $=$= | $0$0 | State the original equation |
| $\left(x^2-4\right)\left(x^2-2\right)$(x2−4)(x2−2) | $=$= | $0$0 | By inspection we can fill in the missing coefficients. |
We can now apply to zero product property to get:
| $x^2-4=0$x2−4=0 | $x^2-2=0$x2−2=0 |
| $x^2=4$x2=4 | $x^2=2$x2=2 |
| $x=2$x=2, $x=-2$x=−2 | $x=\sqrt{2}$x=√2, $x=-\sqrt{2}$x=−√2 |
The solutions are $x=2$x=2, $x=-2$x=−2, $x=\sqrt{2}$x=√2 and $x=-\sqrt{2}$x=−√2.
For the next example we can use technology in order to solve the polynomial.
Example 2
Solve $\frac{x-1}{2}=\frac{\left(x+4\right)^2-22}{3x^2}$x−12=(x+4)2−223x2 using technology.
Think: We need to input the equation into our calculator so that it is in the form $P(x)=0$P(x)=0, where $P(x)$P(x) is the polynomial. Using a TI-84 Plus CE we can -
Do:
- Cross multiply and rearrange the polynomial so it is equal to $3x^3-5x^2-16x+12=0$3x3−5x2−16x+12=0.
- Press the APPS button and select item $9$9: PLYSmlt2.
- Select item $1$1: Polynomial root finder.
- Noting that the polynomial is of degree 3, we set it to order 3.
- We can then press GRAPH, entering the values for $a$a, $b$b, $c$c, and $d$d.
- We can then press F5 to obtain the solutions for $P(x)$P(x).
Therefore the solutions are $x=3$x=3, $x=-2$x=−2, and $x=\frac{2}{3}$x=23.
We can use this function on the calculator for polynomials up to order $10$10.
Practice questions
Question 3
Solve the following equation:
$x^4-16x^2=0$x4−16x2=0
Write all solutions on the same line, separated by commas.
QUESTION 4
Solve the equation $x\left(x-5\right)\left(x+7\right)=8\left(x-5\right)\left(x+7\right)$x(x−5)(x+7)=8(x−5)(x+7).
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.