A real root of a polynomial equation is a real number that, when substituted for the variable, makes the polynomial equal to zero. In other words, it is a solution to the equation where the variable is restricted to the set of real numbers.
Graphically, the real roots are the xx
π₯
-intercepts of the polynomial function's graphβthe points where the curve crosses or touches the xx
π₯
-axis. Polynomials with real coefficients can also have complex roots, which always occur in conjugate pairs and cannot be seen on the real number line. The degree of a polynomial indicates the total number of complex roots, including any real roots counted by their multiplicity.
The fundamental theorem of algebra and root types
The Fundamental Theorem of Algebra guarantees that a polynomial of degree nn
π
has exactly nn
π
complex roots, counting multiplicities. This means that for a polynomial with real coefficients, the roots can be:
-
Real roots: These are real numbers that appear as xx
π₯
-intercepts on the graph. A real root can have a multiplicity greater than one if the graph touches the xx
π₯
-axis and turns back, rather than crossing it.
-
Complex conjugate pairs: Non-real roots of a polynomial with real coefficients always come in pairs of the form aΒ±bia plus or minus b i
πΒ±ππ
, where ii
π
is the imaginary unit (-1the square root of negative 1 end-root
β1β
).
For example, the quadratic equation x2+4=0x squared plus 4 equals 0
π₯2+4=0
has two complex roots, 2i2 i
2π
and -2inegative 2 i
β2π
, and no real roots. Its graph is a parabola that never touches the xx
π₯
-axis. Conversely, x2β4=0x squared minus 4 equals 0
π₯2β4=0
has two real roots, x=2x equals 2
π₯=2
and x=-2x equals negative 2
π₯=β2
, because it can be factored as (xβ2)(x+2)=0open paren x minus 2 close paren open paren x plus 2 close paren equals 0
(π₯β2)(π₯+2)=0
.
Methods for finding real roots
Finding the real roots of a polynomial can range from straightforward for linear and quadratic equations to highly complex for higher degrees.
1. Factoring
This is often the most direct method.
-
Linear equations: A polynomial of degree 1, such as ax+b=0a x plus b equals 0
ππ₯+π=0
, has a single real root at x=βb/ax equals negative b / a
π₯=βπ/π
.
-
Quadratic equations: A degree 2 polynomial, ax2+bx+c=0a x squared plus b x plus c equals 0
ππ₯2+ππ₯+π=0
, can be solved by factoring, if possible, or by using the quadratic formula: x=βbΒ±b2β4ac2ax equals the fraction with numerator negative b plus or minus the square root of b squared minus 4 a c end-root and denominator 2 a end-fraction
π₯=βπΒ±π2β4ππβ2π
. The discriminant, b2β4acb squared minus 4 a c
π2β4ππ
, determines the nature of the roots:
-
If b2β4ac>0b squared minus 4 a c is greater than 0
π2β4ππ>0
, there are two distinct real roots.
-
If b2β4ac=0b squared minus 4 a c equals 0
π2β4ππ=0
, there is exactly one real root with a multiplicity of 2.
-
If b2β4ac<0b squared minus 4 a c is less than 0
π2β4ππ<0
, there are two complex roots and no real roots.
-
-
Higher-degree polynomials: For polynomials of degree 3 or more, factoring can be difficult but is often the best approach once a root has been found using other methods.
2. Rational Root Theorem
This theorem provides a systematic way to test for rational roots of a polynomial with integer coefficients.
-
For a polynomial anxn+β¦+a0=0a sub n x to the n-th power plus β¦ plus a sub 0 equals 0
πππ₯π+β¦+π0=0
, any rational root must be of the form p/qp / q
π/π
, where pp
π
is an integer factor of the constant term a0a sub 0
π0
and qq
π
is an integer factor of the leading coefficient ana sub n
ππ
.
-
You can generate a finite list of possible rational roots, and then test each one using substitution or synthetic division.
-
If a rational root rr
π
is found, the polynomial can be divided by (xβr)open paren x minus r close paren
(π₯βπ)
, yielding a lower-degree polynomial that is easier to solve.
3. Descartes' Rule of Signs
This rule helps predict the number of possible positive and negative real roots, narrowing the search.
-
The number of positive real roots is equal to the number of sign changes in the coefficients of P(x)cap P open paren x close paren
π(π₯)
, or less than that by an even number.
-
The number of negative real roots is found by counting the sign changes in the coefficients of P(βx)cap P open paren negative x close paren
π(βπ₯)
.
4. Graphical methods
Plotting the polynomial function, either by hand or with a graphing calculator, can visually identify the real roots where the graph crosses the xx
π₯
-axis. This is especially useful for approximating roots when analytical methods are too difficult.
5. Numerical methods
For high-degree polynomials or those with complex coefficients, iterative numerical algorithms are used to find approximate roots. These include:
- Newton's Method: An iterative process that refines an initial guess to find a root of any differentiable function, including polynomials.
- Real-root isolation algorithms: Methods based on theorems by Sturm, Budan, and Vincent that find disjoint intervals, each containing exactly one real root.
- Companion matrix methods: A robust technique that finds all complex roots by treating the coefficients of the polynomial as eigenvalues of a special matrix.
Important considerations
-
Multiplicity of roots: A root can appear multiple times. The multiplicity affects the behavior of the graph. At a root with even multiplicity, the graph touches but does not cross the x-axis, while at a root with odd multiplicity, it crosses the x-axis.
-
No real roots: It's important to remember that not all polynomials have real roots. For example, x2+1=0x squared plus 1 equals 0
π₯2+1=0
has complex roots ii
π
and βinegative i
βπ
and no real roots.
-
Numerical stability: High-degree polynomials can be numerically unstable, meaning a small change in coefficients can dramatically affect the roots. For this reason, computational methods need to be carefully designed.