Solve cubic equations using factorisation or other appropriate methods
Factors of Polynomials
Welcome to the world of polynomial factorisation! 🎉 Think of a polynomial as a big Lego set. Factorising is like finding the smaller Lego blocks that fit together to build the whole set. When you can break a polynomial into simpler pieces, solving equations becomes a lot easier.
What is a Factor?
A factor of a polynomial is a polynomial that divides it without leaving a remainder. If $P(x)$ can be written as $P(x)=A(x)\,B(x)$, then $A(x)$ and $B(x)$ are factors of $P(x)$.
Why Factorising Matters
Factorising turns a complicated equation into a product of simpler ones. For example, solving $x^2-5x+6=0$ is trivial once you recognise it as $(x-2)(x-3)=0$. Each factor gives a root, so you get $x=2$ or $x=3$. 🎯
Common Factorisation Techniques
- Greatest Common Factor (GCF) – pull out the biggest common factor from all terms.
- Difference of Squares – $a^2-b^2=(a-b)(a+b)$.
- Sum/Difference of Cubes – $a^3\pm b^3=(a\pm b)(a^2\mp ab+b^2)$.
- Grouping – group terms to factor out common factors.
Solving Cubic Equations
A cubic equation has the form $ax^3+bx^2+cx+d=0$. We’ll use three main steps:
Step 1: Find Rational Roots
Use the Rational Root Theorem: any rational root $p/q$ (in lowest terms) must satisfy $p$ divides $d$ and $q$ divides $a$. Test these candidates in the equation.
Step 2: Factor Out a Linear Term
Once you find a root $r$, factor out $(x-r)$ using polynomial division or synthetic division.
Step 3: Solve the Remaining Quadratic
The cubic now looks like $(x-r)(Ax^2+Bx+C)=0$. Use the quadratic formula $x=\frac{-B\pm\sqrt{B^2-4AC}}{2A}$ to find the other two roots.
Example Problem
Let’s solve $x^3-6x^2+11x-6=0$.
| Step | Details |
|---|---|
| 1. Rational Roots | Possible roots: ±1, ±2, ±3, ±6. Test: $f(1)=0$ → 1 is a root. |
| 2. Factor Out (x-1) | Using synthetic division: $x^3-6x^2+11x-6=(x-1)(x^2-5x+6)$. |
| 3. Solve Quadratic | $x^2-5x+6=0$ factors to $(x-2)(x-3)=0$. Roots: $x=2$ and $x=3$. |
| Result | $x=1,\;2,\;3$. |
Exam Tips
- ?? Check all possible rational roots before moving on.
- ?? Use synthetic division – it’s faster than long division.
- ?? Verify each root by substitution to avoid mistakes.
- ?? Remember the quadratic formula for the remaining quadratic factor.
- 🚀 Practice with different cubic forms to build confidence.
Revision
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