higher-degree inequalities are polynomial inequalities involving a polynomial of degree at least \(3\). Solving them means determining the values of the variable for which the polynomial is positive, negative, non-negative, or non-positive.
Unlike linear or quadratic inequalities, there is no immediate general formula that yields the solution in a single step. The problem rests instead on the analysis of the sign of a product of factors.
For this reason, the key step almost always consists of factoring the polynomial:
\[ \text{factor the polynomial} \]
Once the factorisation is obtained, the inequality reduces to studying the sign of each individual factor and constructing the sign chart.
Contents
- What is a higher-degree polynomial inequality
- General form
- The fundamental principle of sign analysis
- General method of solution
- Multiplicity of roots and sign changes
- Factorable inequalities
- Inequalities with repeated roots
- Inequalities with quadratic factors
- Odd-degree inequalities
- Even-degree inequalities
- The sign chart method
- Common mistakes
- Worked examples
What is a higher-degree polynomial inequality
A higher-degree polynomial inequality is an inequality of the form:
\[ P(x)>0, \qquad P(x)\geq0, \qquad P(x)<0, \qquad P(x)\leq0 \]
where \(P(x)\) is a polynomial of degree at least \(3\).
For example:
\[ x^3-4x>0 \]
or:
\[ x^4-5x^2+4\leq0 \]
or again:
\[ x^5-2x^4-3x^3\geq0. \]
In all these cases the problem consists of determining the intervals of the real line on which the polynomial has the required sign.
General form
A polynomial inequality of degree \(n\) can be written in the form:
\[ a_nx^n+a_{n-1}x^{n-1}+\dots+a_1x+a_0 \gtrless 0 \]
with:
\[ a_n\neq0. \]
The degree of the inequality coincides with the degree of the polynomial.
For example:
\[ x^5-3x^2+1>0 \]
is a fifth-degree inequality.
The fundamental principle of sign analysis
The underlying principle is the following:
the sign of a product depends on the signs of its factors.
Consequently, solving a polynomial inequality requires understanding how the sign of each individual factor varies across the different intervals of the real line.
For example:
\[ (x-2)(x+1)>0 \]
holds when:
- both factors are positive;
- or both factors are negative.
For this reason, the general strategy consists of:
- factoring the polynomial;
- determining the sign of each factor;
- combining the resulting signs.
General method of solution
The standard procedure for solving a higher-degree inequality consists of five key steps.
1. Move everything to one side
The inequality must be written in the form:
\[ P(x)\gtrless0. \]
2. Factor the polynomial
The polynomial is factored as:
\[ P(x)=a(x-x_1)(x-x_2)\dots \]
The main techniques are:
- factoring out a common factor;
- special product identities;
- Ruffini's rule (synthetic division);
- finding rational roots;
- substitution.
3. Find the zeros
The values that make each factor vanish are determined.
4. Construct the sign chart
The zeros partition the real line into intervals. On each interval the sign of the polynomial remains constant.
5. Select the required intervals
Finally, the intervals on which the polynomial satisfies the given inequality are selected.
Multiplicity of roots and sign changes
A key aspect in the study of polynomial inequalities concerns the multiplicity of roots.
Consider:
\[ (x-1)^2. \]
The root:
\[ x=1 \]
has multiplicity \(2\).
In this case the sign of the polynomial does not change as one crosses the root.
Indeed:
\[ (x-1)^2\geq0 \]
on both sides of \(1\).
By contrast, a root of odd multiplicity produces a sign change.
For example:
\[ (x-1)^3 \]
changes sign as one crosses:
\[ x=1. \]
In general:
- a root of even multiplicity \(\Rightarrow\) the sign does not change;
- a root of odd multiplicity \(\Rightarrow\) the sign changes.
Factorable inequalities
Consider the inequality:
\[ x^3-4x>0. \]
Factorisation
Factor out \(x\):
\[ x(x^2-4)>0. \]
Now factor the difference of squares:
\[ x(x-2)(x+2)>0. \]
Sign analysis
The zeros of the polynomial are:
\[ -2,\qquad0,\qquad2. \]
The sign chart is as follows:
| Interval | Sign |
|---|---|
| \((-\infty,-2)\) | \(-\) |
| \((-2,0)\) | \(+\) |
| \((0,2)\) | \(-\) |
| \((2,+\infty)\) | \(+\) |
Since we require:
\[ x(x-2)(x+2)>0, \]
the solution is:
\[ (-2,0)\cup(2,+\infty). \]
Inequalities with repeated roots
Consider:
\[ (x-1)^2(x+3)\geq0. \]
The zeros are:
\[ x=1 \]
with multiplicity \(2\), and:
\[ x=-3 \]
with multiplicity \(1\).
Crossing \(x=-3\) the sign changes, whereas crossing \(x=1\) the sign remains the same.
The sign chart is therefore:
| Interval | Sign |
|---|---|
| \((-\infty,-3)\) | \(-\) |
| \((-3,1)\) | \(+\) |
| \((1,+\infty)\) | \(+\) |
Since the inequality is:
\[ (x-1)^2(x+3)\geq0, \]
the zeros are included in the solution:
\[ [-3,+\infty). \]
Inequalities with quadratic factors
Not all factors of a polynomial need be linear.
Consider:
\[ (x^2-4)(x^2+1)>0. \]
The second factor:
\[ x^2+1 \]
is strictly positive for every real number:
\[ x^2+1>0 \qquad \forall x\in\mathbb{R}. \]
Consequently, the sign of the inequality depends entirely on the factor:
\[ x^2-4. \]
Factoring:
\[ x^2-4=(x-2)(x+2). \]
The inequality therefore reduces to:
\[ (x-2)(x+2)>0, \]
whose solution is:
\[ (-\infty,-2)\cup(2,+\infty). \]
Odd-degree inequalities
Polynomials of odd degree with positive leading coefficient satisfy:
\[ P(x)\to-\infty \qquad \text{as} \qquad x\to-\infty \]
and:
\[ P(x)\to+\infty \qquad \text{as} \qquad x\to+\infty. \]
This end behaviour often makes it possible to predict the overall sign of the polynomial.
For example:
\[ x^3-1 \]
is negative to the left of the root \(x=1\) and positive to the right.
Even-degree inequalities
Polynomials of even degree with positive leading coefficient satisfy instead:
\[ P(x)\to+\infty \]
both as:
\[ x\to-\infty \]
and as:
\[ x\to+\infty. \]
This explains why the sign chart of such polynomials tends to begin and end with the same sign.
The sign chart method
The sign chart is the central tool for solving polynomial inequalities.
The procedure consists of:
- ordering the zeros of the polynomial;
- partitioning the real line into the corresponding intervals;
- determining the sign of each individual factor;
- multiplying the resulting signs together.
It is important to bear in mind that:
- a root of odd multiplicity produces a sign reversal;
- a root of even multiplicity does not produce a sign change.
Common mistakes
Incomplete factorisation
Many errors arise from failing to factor the polynomial completely.
Ignoring the multiplicity of roots
A repeated root does not produce a sign reversal.
Incorrect signs on intervals
It is always advisable to verify the sign by testing a value from each interval.
Incorrectly including the zeros
In strict inequalities:
\[ >,\qquad< \]
the zeros do not belong to the solution set.
In non-strict inequalities:
\[ \geq,\qquad\leq \]
the zeros must be included.
Worked examples
Example 1. Solve:
\[ x^3-x^2-6x>0. \]
Factorisation
Factor out \(x\):
\[ x(x^2-x-6)>0. \]
Factor the trinomial:
\[ x(x-3)(x+2)>0. \]
Sign analysis
The zeros are:
\[ -2,\qquad0,\qquad3. \]
The sign chart gives:
\[ (-2,0)\cup(3,+\infty). \]
Example 2. Solve:
\[ x^4-5x^2+4\leq0. \]
Substitution
Let:
\[ y=x^2. \]
The inequality becomes:
\[ y^2-5y+4\leq0. \]
Factoring:
\[ (y-1)(y-4)\leq0. \]
Hence:
\[ 1\leq y\leq4. \]
Substituting back:
\[ 1\leq x^2\leq4. \]
The solution is:
\[ x\in[-2,-1]\cup[1,2]. \]
Higher-degree polynomial inequalities are solved through the sign analysis of polynomials.
The key idea is to rewrite the polynomial as a product of factors and analyse the sign behaviour across the different intervals of the real line.
The following tools are therefore essential:
- polynomial factorisation;
- finding the roots;
- multiplicity of zeros;
- the sign chart.
Once these tools are understood, even seemingly complex inequalities can be approached in a systematic, rigorous, and well-organised manner.