Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Looking for algebra 2 polynomial functions? Then this Algebra 2 lesson on Polynomial Functions: Zeros, Ends, and Graphs explains each idea step by step, with worked examples.
Quick answer: A polynomial function’s degree and leading coefficient set its end behavior, and the zeros of its factors are the x-intercepts. Odd multiplicity makes the graph cross the axis, and even multiplicity makes it touch and turn. A degree n polynomial has at most n real zeros and at most n − 1 turning points.
Standards: F-IF.C.7c, A-APR.B.3, F-IF.B.4 · Level: Grade 11 Algebra 2
How do the degree and leading coefficient control the graph?
A polynomial function in standard form is f(x) = anxn + … + a1x + a0. The highest power n is the degree and an is the leading coefficient. Together they fix the end behavior, because the leading term dominates when |x| is large.
A polynomial of degree n has at most n real zeros and at most n − 1 turning points.
What do zeros and multiplicity tell you?
A zero of f is a number r with f(r) = 0, where the graph meets the x-axis. If (x − r)k is a factor, then r has multiplicity k. An odd multiplicity makes the graph cross the axis; an even multiplicity makes it touch and turn back. Higher multiplicity flattens the graph at the zero.
| Factor | Zero | Multiplicity | Graph at the zero |
|---|---|---|---|
| (x + 3) | −3 | 1 | Crosses straight through |
| (x − 1)2 | 1 | 2 | Touches and turns |
| (x − 4)3 | 4 | 3 | Crosses, flattened |
How do you sketch a polynomial from its factors?
- Find the zeros and their multiplicities from the factored form.
- Use the degree and leading coefficient to set the end behavior.
- Find the y-intercept, f(0).
- Draw a smooth curve that crosses or touches at each zero as the multiplicity requires.
For f(x) = (x + 3)(x − 1)2, the degree is 3 and the leading coefficient is 1, so the left end falls and the right end rises. The y-intercept is f(0) = 3 · 1 = 3.
A zero of even multiplicity does not change the sign of f. Do not draw the graph crossing at a squared factor, and remember that turning points are not always zeros.
Test one point between zeros to confirm the sign, then use the end behavior as a quick check on the first and last pieces.
- The degree is the highest exponent; the leading coefficient is the coefficient of that term.
- Even degree: both ends point the same way. Odd degree: the ends point opposite ways.
- A positive leading coefficient makes the right end rise; a negative one makes it fall.
- Odd multiplicity: the graph crosses at the zero. Even multiplicity: it touches and turns.
- A degree n polynomial has at most n zeros and at most n − 1 turning points.
Polynomial Functions: Zeros, Ends, and Graphs: frequently asked questions
How do you find the degree of a polynomial?
Write the polynomial in standard form and find the highest exponent. If the polynomial is given as a product of factors, add the degrees of all the factors, counting repeated factors each time.
What does multiplicity mean for a graph?
Multiplicity is the number of times a factor appears. An odd multiplicity makes the graph cross the x-axis, while an even multiplicity makes it touch the axis and turn around. Larger multiplicities flatten the graph near the zero.
How many turning points can a polynomial have?
A polynomial of degree n has at most n − 1 turning points. A cubic can have 0 or 2, and a quartic can have 1 or 3.
Does every polynomial have real zeros?
No. An even-degree polynomial such as x4 + 1 never reaches 0. An odd-degree polynomial always crosses the x-axis at least once because its ends point in opposite directions.
How can polynomials model real situations?
A volume that depends on a cut length, such as an open box made from a sheet, is a cubic function. The domain comes from the situation, and the graph helps find the maximum volume.
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