A higher-degree equation is a polynomial equation whose degree is greater than or equal to \(3\). Unlike first- and second-degree equations, there is no single technique that always yields the solutions directly. The approach depends on the structure of the polynomial and on one's ability to rewrite the equation as a product of simpler factors.
The key idea is that a product equals zero if and only if at least one of its factors equals zero. For this reason, the theory of higher-degree equations revolves largely around polynomial factoring.
In many cases the equation is not solved "directly"; instead, the polynomial is rewritten as a product of lower-degree factors. Once this factored form is obtained, the original equation splits into simpler equations, usually of first or second degree.
Contents
- What is a higher-degree equation
- The zero product property
- Equations solved by factoring out a common factor
- Equations solved using special factoring formulas
- Equations factored by synthetic division (Ruffini's rule)
- Biquadratic equations
- Trinomial-form equations
- Multiplicity of roots
- General strategy for solving higher-degree equations
- Common mistakes to avoid
What is a higher-degree equation
A higher-degree equation is a polynomial equation of the form:
\[ P(x)=0 \]
where \(P(x)\) is a polynomial of degree greater than or equal to \(3\).
For example:
\[ x^3-4x=0, \]
\[ x^4-5x^2+4=0, \]
\[ x^5-2x^4+x^2=0 \]
are all higher-degree equations.
The degree of the equation equals the highest power of the variable after all terms have been collected.
For example, in the equation:
\[ x^4-3x^2+1=0 \]
the degree is \(4\), since the highest power of \(x\) is \(4\).
The zero product property
The fundamental property used in solving higher-degree equations is the zero product property:
\[ A\cdot B=0 \quad \Longleftrightarrow \quad A=0 \ \text{or} \ B=0. \]
More generally:
\[ A_1\cdot A_2\cdot \dots \cdot A_n=0 \]
if and only if at least one of the factors is zero.
This property is the cornerstone of the entire theory. Once the polynomial has been factored, the original equation becomes a product equal to zero.
Example
Solve:
\[ x^3-4x=0. \]
Factor out the common factor \(x\):
\[ x(x^2-4)=0. \]
The binomial \(x^2-4\) is a difference of squares:
\[ x^2-4=(x-2)(x+2). \]
Thus:
\[ x(x-2)(x+2)=0. \]
Applying the zero product property:
\[ x=0 \]
or:
\[ x-2=0 \]
or:
\[ x+2=0. \]
The solutions are:
\[ x=0,\qquad x=2,\qquad x=-2. \]
Equations solved by factoring out a common factor
In many higher-degree equations, the first step is to factor out a common factor.
Example
Solve:
\[ x^4-3x^3=0. \]
Every term contains the factor \(x^3\). Factoring it out:
\[ x^3(x-3)=0. \]
Applying the zero product property:
\[ x^3=0 \]
or:
\[ x-3=0. \]
The first equation gives:
\[ x=0, \]
while the second gives:
\[ x=3. \]
The solution set is:
\[ S=\{0,3\}. \]
Factoring out a common factor is often the quickest method and should always be the first thing to check.
Equations solved using special factoring formulas
Many equations can be factored by applying standard algebraic identities.
Example
Solve:
\[ x^4-16=0. \]
Notice that:
\[ 16=4^2, \]
so:
\[ x^4-16=(x^2)^2-4^2. \]
Applying the difference of squares identity:
\[ x^4-16=(x^2-4)(x^2+4). \]
The first factor can be factored further:
\[ x^2-4=(x-2)(x+2). \]
We obtain:
\[ (x-2)(x+2)(x^2+4)=0. \]
Setting each factor equal to zero:
\[ x-2=0, \]
\[ x+2=0, \]
\[ x^2+4=0. \]
The first two yield:
\[ x=2, \qquad x=-2. \]
The equation:
\[ x^2+4=0 \]
has no real solutions, since:
\[ x^2=-4 \]
is impossible in the real numbers.
Therefore:
\[ S=\{-2,2\}. \]
Equations factored by synthetic division (Ruffini's rule)
When the polynomial cannot be factored immediately by inspection, it is useful to look for rational roots and apply Ruffini's rule (equivalently, synthetic division).
Example
Solve:
\[ x^3-6x^2+11x-6=0. \]
We test the integer divisors of the constant term \(6\):
\[ \pm1,\quad \pm2,\quad \pm3,\quad \pm6. \]
Substituting \(x=1\):
\[ 1-6+11-6=0. \]
So \(x=1\) is a root of the polynomial.
We may therefore divide the polynomial by \(x-1\) using Ruffini's rule, obtaining:
\[ x^3-6x^2+11x-6=(x-1)(x^2-5x+6). \]
The remaining trinomial factors as:
\[ x^2-5x+6=(x-2)(x-3). \]
The equation becomes:
\[ (x-1)(x-2)(x-3)=0. \]
The solutions are:
\[ x=1,\qquad x=2,\qquad x=3. \]
Biquadratic equations
A particularly important special case is that of biquadratic equations, which have the form:
\[ ax^4+bx^2+c=0. \]
These equations involve only \(x^4\), \(x^2\), and the constant term.
The key idea is to introduce the substitution:
\[ y=x^2, \]
which reduces the equation to a quadratic in \(y\).
Example
Solve:
\[ x^4-5x^2+4=0. \]
Let:
\[ y=x^2. \]
The equation becomes:
\[ y^2-5y+4=0. \]
Factoring:
\[ (y-1)(y-4)=0. \]
Thus:
\[ y=1 \]
or:
\[ y=4. \]
Returning to the original variable:
\[ x^2=1 \]
or:
\[ x^2=4. \]
Solving:
\[ x=\pm1, \qquad x=\pm2. \]
The solution set is:
\[ S=\{-2,-1,1,2\}. \]
Trinomial-form equations
Some higher-degree equations have a structure analogous to quadratic equations.
For example:
\[ x^6-5x^3+6=0. \]
Here we set:
\[ y=x^3. \]
The equation becomes:
\[ y^2-5y+6=0. \]
Factoring:
\[ (y-2)(y-3)=0. \]
Thus:
\[ y=2 \]
or:
\[ y=3. \]
Going back to the original variable:
\[ x^3=2 \]
or:
\[ x^3=3. \]
The real solutions are:
\[ x=\sqrt[3]{2}, \qquad x=\sqrt[3]{3}. \]
Multiplicity of roots
A root may appear more than once in the factored form of the polynomial.
For example:
\[ (x-2)^3(x+1)=0. \]
The solutions are:
\[ x=2 \]
and:
\[ x=-1. \]
However, \(x=2\) appears three times in the factored form, so it is called a root of multiplicity three (or a triple root).
The multiplicity of a root is especially significant in the study of polynomial functions and their graphs.
General strategy for solving higher-degree equations
In practice, it is best to follow a systematic approach.
- Move all terms to one side of the equation.
- Factor out any common factors.
- Look for applicable special factoring formulas.
- Consider a suitable substitution.
- Apply Ruffini's rule (synthetic division) if needed.
- Factor the polynomial completely.
- Apply the zero product property to each factor.
The ultimate goal is always the same: rewrite the equation as a product of factors set equal to zero.
Common mistakes to avoid
The first mistake is forgetting that the property:
\[ AB=0 \quad \Longrightarrow \quad A=0 \ \text{or} \ B=0 \]
holds only when the product equals zero.
For example:
\[ AB=6 \]
does not imply:
\[ A=6 \qquad \text{or} \qquad B=6. \]
The second mistake is stopping the factoring process too early. For instance:
\[ x^4-16=(x^2-4)(x^2+4) \]
is not yet fully factored, since:
\[ x^2-4=(x-2)(x+2). \]
The third mistake is losing solutions during substitution. When one sets:
\[ y=x^2, \]
it is essential to remember that:
\[ x^2=4 \]
yields two solutions:
\[ x=2 \qquad \text{and} \qquad x=-2. \]
Higher-degree equations cannot be solved by a single universal formula; they require factoring techniques and polynomial transformations.
The central principle is always the same: reduce the equation to a product of factors set equal to zero, then apply the zero product property.
For this reason, a solid command of factoring, special algebraic identities, and Ruffini's rule is indispensable. Higher-degree equations are indeed a meeting point of elementary algebra, polynomial theory, and the study of functions.