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Solve a cubic inequality

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Solving cubic inequalities

First write the inequality in zero form and find the ordered roots with their multiplicities. Track the sign from one end of the cubic, changing it only at odd-multiplicity roots, then apply the endpoint rule

\[F(x)=a\prod_i\left(x-r_i\right)^{m_i}\quad\text{and compare }F(x)\text{ with }0\]
  1. 1Write zero form Move all terms to one side so the polynomial is compared with zero \(\displaystyle F(x)>4\Rightarrow F(x)-4>0\)
  2. 2Find ordered roots Find every distinct critical root and record the multiplicity of each factor \(\displaystyle (x+2)^2(x-1)=0\Rightarrow r_1=-2\ (m=2),\ r_2=1\ (m=1)\)
  3. 3Start with the leading sign For a cubic, the far-left sign is opposite to the sign of the leading coefficient \(\displaystyle a>0\Rightarrow F(x)<0\text{ on the far-left interval}\)
  4. 4Change sign only at odd roots An odd-multiplicity root changes the interval sign, while an even-multiplicity root keeps it the same \(\displaystyle (x+2)^2\text{ keeps the sign across }x=-2,\qquad(x-1)\text{ changes it}\)
Year 11, VCE Units 1 & 2, Chapter 4, 4 difficulty levels, No calculatorBuild a worksheet

Solving cubic inequalities

First write the inequality in zero form and find the ordered roots with their multiplicities. Track the sign from one end of the cubic, changing it only at odd-multiplicity roots, then apply the endpoint rule

\[F(x)=a\prod_i\left(x-r_i\right)^{m_i}\quad\text{and compare }F(x)\text{ with }0\]
  1. Write zero form Move all terms to one side so the polynomial is compared with zero \(\displaystyle F(x)>4\Rightarrow F(x)-4>0\)
  2. Find ordered roots Find every distinct critical root and record the multiplicity of each factor \(\displaystyle (x+2)^2(x-1)=0\Rightarrow r_1=-2\ (m=2),\ r_2=1\ (m=1)\)
  3. Start with the leading sign For a cubic, the far-left sign is opposite to the sign of the leading coefficient \(\displaystyle a>0\Rightarrow F(x)<0\text{ on the far-left interval}\)
  4. Change sign only at odd roots An odd-multiplicity root changes the interval sign, while an even-multiplicity root keeps it the same \(\displaystyle (x+2)^2\text{ keeps the sign across }x=-2,\qquad(x-1)\text{ changes it}\)

Worked example

Solve the cubic inequality and give the solution set

\[-(x-2)(x+1)^2>0\]
  1. \(\displaystyle r_1=-1\ (m=2),\qquad r_2=2\ (m=1)\)The double root is -1 and the simple root is 2.
  2. \(\displaystyle a=-1<0\Rightarrow F(x)>0\text{ on the far-left interval}\)A cubic with a negative leading coefficient is positive on the far-left end
  3. \(\displaystyle \text{The sign stays positive across }x=-1\text{ and changes to negative after }x=2\)The double root does not change the sign, while the simple root does
  4. \(\displaystyle x\in\left(-\infty,-1\right)\cup\left(-1,2\right)\)The relation is strict, so both roots are excluded from the positive intervals

Common mistakes

Changing at an even root
✕\(\displaystyle (x+1)^2:\quad\text{the sign changes across }x=-1\)✓\(\displaystyle (x+1)^2:\quad\text{the sign stays the same across }x=-1\)
An even-multiplicity factor has the same sign on both sides of its root
Using the wrong endpoint rule
✕\(\displaystyle F(r)=0:\quad r\text{ is excluded from the solution of }F(x)\ge0\)✓\(\displaystyle F(r)=0:\quad r\text{ is included in the solution of }F(x)\ge0\)
An inclusive inequality includes a root where the polynomial value is zero
Reversing the far-left sign
✕\(\displaystyle F(x)=-x^3\text{ is negative for sufficiently small }x\)✓\(\displaystyle F(x)=-x^3\text{ is positive for sufficiently small }x\)
An odd-degree polynomial with a negative leading coefficient is positive on the far-left end

Difficulty levels

introSolve a one-root monotone cubic inequality from factorised information
standardComplete a sign chart for a cubic with three simple roots
challengeHandle a repeated root, a negative leading scale or an inequality requiring rearrangement
examFactorise, complete interval signs and select a strict or inclusive solution set

Questions about this topic

Why does an even-multiplicity root keep the sign unchanged?+

The corresponding factor is positive on both sides of the root, so multiplying by it does not reverse the sign when the root is crossed

How is the far-left sign found for a cubic?+

For an odd-degree polynomial, the far-left sign is opposite to the sign of its leading coefficient

When are roots included in a cubic inequality solution?+

Include a root for an inclusive sign such as ≥ or ≤. Exclude it for a strict sign such as > or <

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