A polynomial is an expression made of terms ( a_kx^k ) with whole-number exponents, such as ( 3x^4 – 5x^2 + x – 8 ). Its degree is the highest power (here 4), and its zeros are the x-values where it equals 0. A degree-n polynomial has at most n real zeros, and ( x = c ) is a zero exactly when ( (x – c) ) is a factor.
A polynomial is an expression built from a variable and constants using only addition, subtraction, multiplication, and whole-number powers:
$$p(x) = a_nx^n + a_{n-1}x^{n-1} + dots + a_1x + a_0$$
The numbers ( a_n, dots, a_0 ) are the coefficients, and the exponents must be non-negative integers. So ( 4x^3 – 2x + 7 ) is a polynomial, but ( x^{-1} ), ( sqrt{x} ) and ( frac{1}{x + 1} ) are not.
Polynomial vocabulary
| Term | Meaning | In ( 3x^4 – 5x^2 + x – 8 ) |
|---|---|---|
| Term | one piece, ( a_kx^k ) | ( 3x^4 ), ( -5x^2 ), ( x ), ( -8 ) |
| Degree | highest power | 4 |
| Leading term | term with the highest power | ( 3x^4 ) |
| Leading coefficient | its coefficient | 3 |
| Constant term | term with no x | ( -8 ) |
Polynomials are named by degree:
| Degree | Name | Example |
|---|---|---|
| 0 | constant | ( 5 ) |
| 1 | linear | ( 2x + 1 ) |
| 2 | quadratic | ( x^2 – 3x + 2 ) |
| 3 | cubic | ( x^3 – 6x^2 + 11x – 6 ) |
| 4 | quartic | ( x^4 – 5x^2 + 4 ) |
and by number of terms: monomial (1), binomial (2), trinomial (3).
Zeros (roots) of a polynomial
A zero of ( p ) is an x-value where ( p(x) = 0 ). On the graph, real zeros are the x-intercepts. The factor theorem links zeros and factors:
$$p(c) = 0 iff (x - c) text{ is a factor of } p(x)$$
So once you factor a polynomial, its zeros can be read off.
Example 1: factoring out a common factor
( p(x) = 2x^3 – 8x = 2x(x^2 – 4) = 2x(x – 2)(x + 2) ). Zeros: ( 0, 2, -2 ).
Example 2: factoring by grouping
( p(x) = x^3 – x^2 – 4x + 4 = x^2(x – 1) – 4(x – 1) = (x – 1)(x^2 – 4) ). Zeros: ( 1, 2, -2 ).
Example 3: quadratic in form
( p(x) = x^4 – 5x^2 + 4 ). Let ( u = x^2 ): ( u^2 – 5u + 4 = (u – 1)(u – 4) ). As a result, ( (x^2 – 1)(x^2 – 4) = (x – 1)(x + 1)(x – 2)(x + 2) ), and the zeros are ( pm1, pm2 ).
The rational root test
If a polynomial with integer coefficients has a rational zero ( frac{p}{q} ) in lowest terms, then p divides the constant term and q divides the leading coefficient.
Example 4
Find the zeros of ( p(x) = x^3 – 6x^2 + 11x – 6 ).
Candidates. The leading coefficient is 1, so the candidates are the divisors of 6: ( pm1, pm2, pm3, pm6 ).
Test. ( p(1) = 1 – 6 + 11 – 6 = 0 ). So ( x = 1 ) is a zero.
Divide out ( (x – 1) ) with synthetic division:
| 1 | −6 | 11 | −6 | |
|---|---|---|---|---|
| 1 | 1 | −5 | 6 | |
| 1 | −5 | 6 | 0 |
The bottom row gives the quotient ( x^2 – 5x + 6 ) with remainder 0. Factor it: ( (x – 2)(x – 3) ). So ( p(x) = (x – 1)(x – 2)(x – 3) ), with zeros 1, 2, 3.
When the leftover quadratic doesn’t factor, finish with the quadratic formula.
Multiplicity
If ( (x – c)^k ) is a factor, then c is a zero of multiplicity k. The multiplicity shows on the graph:
- Odd multiplicity (1, 3, …): the graph crosses the x-axis. Multiplicity 3, for example, flattens it as it crosses.
- Even multiplicity (2, 4, …): the graph touches the axis and turns back.
Example 5. ( p(x) = -2x^4 + 8x^2 = -2x^2(x – 2)(x + 2) ). The zero ( x = 0 ) has multiplicity 2, so the graph touches the origin without crossing; it crosses at ( pm2 ).
The fundamental theorem of algebra
A polynomial of degree n has exactly n zeros, counting multiplicity and complex zeros. Complex zeros of real polynomials come in conjugate pairs ( a pm bi ). So a cubic always has at least one real zero, and ( x^2 + 1 ) has the two complex zeros ( pm i ).
End behavior
For large ( |x| ), the leading term dominates, so the degree and the sign of the leading coefficient decide what the graph does at the far left and right.
| Leading coefficient ( > 0 ) | Leading coefficient ( < 0 ) | |
|---|---|---|
| Even degree | ↑ on both ends | ↓ on both ends |
| Odd degree | ↓ left, ↑ right | ↑ left, ↓ right |
A degree-n polynomial also has at most ( n – 1 ) turning points, because its derivative has degree ( n – 1 ).
Polynomial roots calculator
Enter any polynomial to get its real zeros, degree, leading coefficient, turning points and end behavior.
The full-page polynomial roots calculator has more examples.
Sketching a polynomial graph
- Factor to find the zeros and their multiplicities.
- Find the y-intercept, ( p(0) ).
- Use the end behavior for the far left and right.
- Join the pieces, crossing or touching at each zero.
Example 6. ( p(x) = (x + 1)^2(x – 3) ): zeros ( -1 ) (touch) and 3 (cross); y-intercept ( p(0) = -3 ); degree 3 with positive leading coefficient, so it starts low and ends high. Starting from the bottom left, it rises to touch the axis at ( -1 ), dips to a minimum below the axis, then crosses at 3 and rises.
Polynomials in calculus
Polynomials are the easiest functions in calculus: they’re continuous everywhere, and the power rule differentiates them term by term. As a result, they are the natural testing ground for critical points and inflection points. Taylor series go the other way, approximating complicated functions by polynomials. Also, Newton’s method finds zeros that factoring can’t.
Common mistakes
- Sign of the factor. A zero at ( x = -2 ) gives the factor ( (x + 2) ).
- Skipping zero coefficients in synthetic division. For ( x^3 – 2x – 5 ), write 1, 0, −2, −5.
- Calling non-polynomials polynomials. Negative or fractional exponents and variables in denominators disqualify an expression.
- Assuming all zeros are real. For example, ( x^3 – 2x – 5 ) has one real zero and two complex ones.
Practice problems
- State the degree and leading coefficient of ( 7 – 2x^5 + x^2 ).
- Find the zeros of ( x^2 – 9x + 20 ).
- Find the zeros of ( x^3 – 4x^2 + x + 6 ) (hint: try ( x = -1 )).
- Find the zeros and multiplicities of ( (x – 4)^3(x + 1)^2 ).
- Describe the end behavior of ( -x^3 + 2x ).
- What is the maximum number of turning points of a degree-6 polynomial?
Answers: (1) degree 5, leading coefficient −2; (2) 4, 5; (3) −1, 2, 3; (4) 4 (multiplicity 3), −1 (multiplicity 2); (5) up on the left, down on the right; (6) 5.
Zeros of polynomials: more questions
- Find the zeros of ( x^3 – 9x ).
- If ( 2 ) is a zero of ( x^3 – 3x^2 + kx – 2 ), find k.
- Write a polynomial with zeros ( -1 ), 2 and 5 and leading coefficient 1.
- Find all zeros of ( x^3 – 2x^2 – 5x + 6 ).
- Find the sum and product of the zeros of ( 2x^2 – 8x + 3 ).
Answers: (1) ( 0, 3, -3 ); (2) ( k = 3 ), since ( 8 – 12 + 2k – 2 = 0 ); (3) ( (x + 1)(x – 2)(x – 5) = x^3 – 6x^2 + 3x + 10 ); (4) ( 1, -2, 3 ); (5) sum 4, product ( frac32 ).
FAQ
What is a polynomial?
An expression made of terms ( a_kx^k ) with whole-number exponents, such as ( 3x^2 – 5x + 1 ).
What is the degree of a polynomial?
The highest power of the variable with a non-zero coefficient.
How do you find the zeros of a polynomial?
Factor it (common factors, grouping, the rational root test with synthetic division), then set each factor equal to zero. Use the quadratic formula or a numerical method for what’s left.
Is a constant a polynomial?
Yes. A non-zero constant like 5 is a polynomial of degree 0.
What is the difference between a zero and a root?
Nothing important: a zero is an input where the polynomial equals 0, and a root is a solution of the equation ( p(x) = 0 ). They’re the same numbers.
Further reading
- Zeroes/Roots of Polynomials (Paul’s Online Math Notes) — multiplicity, the fundamental theorem of algebra and factoring.
- Finding Zeroes of Polynomials (Paul’s Online Math Notes) — the rational root test and synthetic division in practice.
- Polynomial (Wikipedia) — definitions, history and properties.
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