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Move between roots, factors, polynomial rules and equations

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Test the rational candidate x=1 in the polynomial

\[P(1)=1^3-6(1)^2+11(1)-6\]
Year 12, VCE Units 3 & 4, Chapter 12, 4 difficulty levels, No calculatorBuild a worksheet

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

\[P(a)=0\Longleftrightarrow\left(x-a\right)\text{ is a factor of }P(x)\]
  1. Write zero form Move all terms to one side before applying the null factor law \(\displaystyle P(x)=4\Rightarrow P(x)-4=0\)
  2. Test rational candidates Use factors of the constant and leading coefficients to list possible rational roots \(\displaystyle P(x)=2x^3+3x^2-8x-12\Rightarrow x=\frac{p}{q},\quad p\mid12,\ q\mid2\)
  3. Match a root to its factor A root a gives the linear factor x-a. \(\displaystyle x=-3\Rightarrow x+3=0\)
  4. 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

\[x^3-6x^2+11x-6=0\]
  1. \(\displaystyle P(1)=1^3-6(1)^2+11(1)-6=0\)Test a rational candidate and use the zero value to identify a factor
  2. \(\displaystyle P(x)=(x-1)(x^2-5x+6)\)Divide by x-1 to obtain the quadratic quotient
  3. \(\displaystyle x^2-5x+6=(x-2)(x-3)\)Factor the quadratic quotient
  4. \(\displaystyle (x-1)(x-2)(x-3)=0\Rightarrow x=1,\ 2,\ 3\)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 root is the value that makes the factor zero, so x+4=0 gives x=-4.
Stopping 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 continue
Incorrect 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 recorded

Difficulty 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 the factor corresponding to a root occurs. For example, when the factor corresponding to root 2 occurs twice, that root has multiplicity 2

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 12: Continuous probability distributions

In this chapter

12A Continuous random variables and density functions: Validate and normalise a continuous probability density function12A Continuous random variables and density functions: Estimate probabilities from grouped continuous frequency dataGroup numerical data into class intervalsConstruct a histogram from a frequency table12A Continuous random variables and density functions: Finite events, probability assignments and complements12A Continuous random variables and density functions: Conditional probability from exact representationsEvaluate and determine function rules12B Probability density validity and normalisation: Validate and normalise a continuous probability density function12B Probability density validity and normalisation: Estimate probabilities from grouped continuous frequency dataSolve a linear equation12B Probability density validity and normalisation: Evaluate a polynomial or recover a coefficient parameterWork with sets and interval notation12B Probability density validity and normalisation: Round to decimal places and significant figuresInterval and conditional probabilities from a densityCompound events, the addition rule and mutual exclusivity12C Interval and conditional probabilities: Finite events, probability assignments and complements12C Interval and conditional probabilities: Conditional probability from exact representationsMean, variance and standard deviation of a continuous random variableMedian, quartiles and percentiles of a continuous random variableExpand and factor polynomial expressions12D Mean, variance, standard deviation and percentiles: Evaluate a polynomial or recover a coefficient parameterMove between roots, factors, polynomial rules and equationsSolve a quadratic exactly with the quadratic formula12D Mean, variance, standard deviation and percentiles: Round to decimal places and significant figuresLinear transformations and totals of independent continuous random variables