Chapter 3 · Complex Numbers and Equations · Concept 3
3-17 High-Degree Equations
A quadratic can model one peak or one valley. A cubic can model something that starts flat, surges — like a heartbeat — and settles again. Which is why anyone designing a heart valve ends up solving one.
Guiding question
How do cubics let you model a system more truthfully than a quadratic can?
Learning outcome. Solve higher-degree equations.
Degree n means n solutions
An equation of degree n always has n solutions when complex numbers are allowed — counting a repeated solution as many times as it repeats.
Look at how far that has come. In Grade X a quadratic had two solutions, or one, or none, depending on the discriminant. Chapter 3 removed the “none”, and now the “one” is really two that landed on the same spot. Every equation of degree n has exactly n. No exceptions and no special cases.
Step 1 — factorise. Three routes:
cubes
If it is x³ + a³ = 0:
x³ + a³ = (x + a)(x² − ax + a²)
a chunk
If a quartic has only x⁴, x² and a number, treat x² as one chunk and factorise it as a quadratic.
otherwise
Use the Factor Theorem from lesson 3-16 — hunt for a root among ±(divisors of the constant), then divide it out.
Step 2 — read off the solutions.
(x − α)(x − β)(x − γ) = 0 gives x = α or β or γ
And if a quadratic factor will not split, use the quadratic formula on it — which is where the imaginary solutions come from.
Real solutions sit on the flat line — the number line you have always used. Imaginary ones lift off it. A dashed ring means a solution that arrived more than once.
x³ − 8 = 0
the degree
3
solutions, counting repeats
?
(a) Solve x³ − 8 = 0.
- 1
Recognise the shape before doing anything.
x3 − 8 = x3 − 23A difference of two cubes. There is a formula for exactly this, and hunting for roots would be slower.
Work with a partner. Before solving each one, agree on which of the three routes it wants. Getting that right saves more time than doing the algebra faster ever will.
(a) is a cube in disguise.
x³ = −1 is x³ + 1 = 0, which is x³ + 1³ — so it factorises as (x + 1)(x² − x + 1). Move everything to one side before deciding which route to take.
Solve.
Count your answers before you stop.
A cubic needs three and a quartic needs four, repeats included. If you have fewer, you have either stopped at a quadratic factor without solving it, or written x² = 9 and forgotten that it gives two answers. (l) is worth a look: it has one solution written three times, because the whole equation is (x + 2)³ = 0.
Water flows into a tank and out of it, both at rates that depend on the pressure x:
in
x⁴ + x² + x
out
x³ + 2
Write the equation for the level staying perfectly steady, solve it, and say whether the answers are all real.
Handing in too few solutions.
Degree 3 means three, degree 4 means four, repeats counted. Count them before you stop.
Stopping at x² = 9.
That is not a solution, it is a step. It gives x = 3 AND x = −3, and the same goes for x² = −2.
Leaving a quadratic factor unsolved.
(x − 2)(x² − 2x − 4) = 0 has three solutions, not one. The bracket that will not factorise still gets the formula.
Getting the middle sign of the cube formula wrong.
x³ − a³ = (x − a)(x² + ax + a²). The sign inside the long bracket is the opposite of the short one.
Not moving everything to one side first.
x³ = −1 has to become x³ + 1 = 0 before you can see it is a sum of cubes.
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