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By the end of these notes, you should be able to:
A cubic equation is an equation where the highest power of x is 3. It looks like this:
ax3+bx2+cx+d=0
where a, b, c, and d are numbers, and a ≠ 0.
Examples of cubic equations:
A cubic equation can have up to three solutions (also called roots). Your job is to find all of them.
The strategy for solving a cubic equation has three stages:
Stage 1 → Find one linear factor using the Factor Theorem Stage 2 → Divide out that factor to get a quadratic Stage 3 → Solve the quadratic to find the remaining roots
Let's look at each stage carefully.
Recall the Factor Theorem:
If you substitute x = a into f(x) and get f(a) = 0, then (x − a) is a factor of f(x).
How to find that first factor — use "trial and error" with factors of the constant term:
If your cubic is f(x) = ax³ + bx² + cx + d, then your first step is to look at the constant term d and list all of its factors (the numbers that divide into it exactly). Then try substituting them one by one into f(x) until you find one that gives zero.
💡 Why factors of d? Linear factors of a polynomial with integer coefficients will always have roots that are factors of the constant term (or simple fractions involving factors of the leading coefficient). This is where you always start looking.
Example: For factors of 15, you would try: ±1, ±3, ±5, ±15
For non-integer roots: If the leading coefficient (the number in front of x³) is not 1, you should also try simple fractions like ±½, ±¼, etc. formed using factors of the leading coefficient as denominators.
Once you have found a linear factor, say (x − 1), you need to divide it out of the cubic to get a quadratic (a degree-2 expression). There are three methods to do this:
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