Roots, factors and polynomial equations
Write a polynomial equation in zero form, factorise it, and then apply the null factor law. A root gives a linear factor with the opposite constant sign, and repeated factors must be retained
\[P(a)=0\Longleftrightarrow\left(x-a\right)\text{ is a factor of }P(x)\]
- Write zero form Move all terms to one side before applying the null factor law \(\displaystyle P(x)=4\Rightarrow P(x)-4=0\)
- Test rational candidates Use factors of the constant and leading coefficients to list possible rational roots \(\displaystyle P(x)=2x^3+3x^2-8x-12\Rightarrow x=\frac{p}{q},\quad p\mid12,\ q\mid2\)
- Match a root to its factor A root a gives the linear factor x-a. \(\displaystyle x=-3\Rightarrow x+3=0\)
- Keep multiplicities A repeated root contributes the same linear factor more than once \(\displaystyle (x-2)^2(x+1)=0\Rightarrow x=2\text{ twice and }x=-1\)
Worked example
Solve the polynomial equation exactly
\[x^3-6x^2+11x-6=0\]
- \(\displaystyle P(1)=1^3-6(1)^2+11(1)-6=0\)Test a rational candidate and use the zero value to identify a factor
- \(\displaystyle P(x)=(x-1)(x^2-5x+6)\)Divide by the identified linear factor to obtain the quadratic quotient
- \(\displaystyle x^2-5x+6=(x-2)(x-3)\)Factor the quadratic quotient
- \(\displaystyle (x-1)(x-2)(x-3)=0\Rightarrow x=1,\ 2,\ 3\)Apply the null factor law to obtain every solution
Common mistakes
Reversing the factor sign
✕\(\displaystyle x+4=0\Rightarrow x=4\)✓\(\displaystyle x+4=0\Rightarrow x=-4\)
The factor is zero at negative four, not positive fourStopping after one factor
✕\(\displaystyle P(x)=(x-1)(x^2-5x+6)\text{ is completely factorised}\)✓\(\displaystyle P(x)=(x-1)(x-2)(x-3)\)
The quadratic quotient still has factors, so the factorisation must continueIncorrect multiplicity
✕\(\displaystyle (x-2)^2(x+1)=0:\quad x=2\text{ has multiplicity }1\)✓\(\displaystyle (x-2)^2(x+1)=0\Rightarrow x=2\text{ with multiplicity 2, and }x=-1\)
A repeated factor represents the same root more than once and its multiplicity must be recordedDifficulty levels
introSolve an already factorised cubic equation using the null factor law
standardFactorise a monic cubic or construct its expanded rule from stated roots
challengeHandle rational roots, repeated roots, irreducible quadratic factors or quartic structure
examTest a candidate, divide, choose the factor form and state all ordered real solutions
Questions about this topic
What is a root of a polynomial?+
A root is an input a for which P(a)=0. A real root is also the horizontal coordinate of an x-intercept
Why must a polynomial equation be in zero form?+
The null factor law says a product is zero only when at least one factor is zero, so one side must be zero before the law is used
What does multiplicity mean?+
Multiplicity is the number of times a root occurs as a factor. A squared linear factor gives its root multiplicity 2
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 11, VCE Units 1 & 2, Mathematical Methods, chapter 4: Cubic polynomials
In this chapter
Recognise and classify a polynomialEvaluate a polynomial or recover a coefficient parameterAdd, subtract or multiply polynomialsDivide polynomials and state quotient and remainderAnalyse a cubic graph or determine its ruleDetermine parameters in a polynomial identityApply the remainder and factor theoremsMove between roots, factors, polynomial rules and equationsSolve a cubic inequalityIntersections of a cubic with a line or quadraticFormulate and interpret a cubic model