What is a polynomial function?
A polynomial is a sum of terms like a·xⁿ with whole-number powers, such as f(x) = x³ − 2x² − 5x + 6. The degree is the highest power and the leading coefficient is the number in front of it.
End behaviour
Far to the left and right, only the leading term matters:
- Even degree, positive leading coefficient: both ends rise.
- Even degree, negative leading coefficient: both ends fall.
- Odd degree, positive leading coefficient: falls on the left, rises on the right.
- Odd degree, negative leading coefficient: rises on the left, falls on the right.
A polynomial of degree n has at most n real zeros and at most n − 1 turning points.
Zeros, factors and multiplicity
A zero (or root) is an x-value where f(x) = 0, which is where the graph meets the x-axis. The Factor Theorem says: x − c is a factor of f(x) exactly when f(c) = 0.
The Remainder Theorem says that dividing f(x) by x − c leaves remainder f(c). So you can find a remainder just by evaluating.
Multiplicity counts how many times a factor repeats. In f(x) = (x − 1)²(x + 4) the zero x = 1 has multiplicity 2: the graph touches the axis and turns back. A zero of multiplicity 1 is crossed straight through.
Rational zero test
If f has whole-number coefficients, any rational zero p/q has p dividing the constant term and q dividing the leading coefficient. For 2x³ − 5x + 3, p comes from ±1, ±3 and q from ±1, ±2, so candidates include ±1, ±3, ±1/2, ±3/2.
Worked example
Find all zeros of f(x) = x³ − 2x² − 5x + 6.
- Test candidates from ±1, ±2, ±3, ±6. f(1) = 1 − 2 − 5 + 6 = 0, so x − 1 is a factor.
- Divide by x − 1 (long or synthetic division) to get x² − x − 6.
- Factor: x² − x − 6 = (x − 3)(x + 2).
- So f(x) = (x − 1)(x − 3)(x + 2) and the zeros are 1, 3 and −2.
The y-intercept is f(0) = 6, which matches the product of the constants: (−1)(−3)(2) = 6.
Sketching
Plot the zeros, mark the y-intercept, decide crossing or touching from multiplicity, and make the ends match the end behaviour. Here the degree is 3 with a positive leading coefficient, so the graph rises on the right and falls on the left.
Common mistakes
- Using f(−c) instead of f(c) for the factor x − c. (For x + 3, test c = −3.)
- Forgetting that missing terms need a zero placeholder in synthetic division.
- Assuming a degree-n polynomial always has n real zeros; some are complex.