Math · College algebra · Concept
Polynomial functions: zeros and end behavior
A polynomial function is a sum of terms aₙxⁿ + … + a₁x + a₀ with whole-number powers. Its degree n and leading coefficient aₙ decide its end behavior, the direction of the graph far to the left and right. Its real zeros are its x-intercepts: at a zero of odd multiplicity the graph crosses the x-axis, and at a zero of even multiplicity it touches the axis and turns back. A polynomial of degree n has at most n real zeros and at most n − 1 turning points.
Degree and leading coefficient
The degree is the highest power of x, and the leading coefficient is the number multiplying it. For large |x| the leading term outweighs all the others, so it alone decides where the graph heads.
End behavior
Four cases cover every polynomial.
| Degree | Leading coefficient | Left end | Right end |
|---|---|---|---|
| Even | Positive | Up | Up |
| Even | Negative | Down | Down |
| Odd | Positive | Down | Up |
| Odd | Negative | Up | Down |
Zeros and multiplicity
If (x − r)ᵏ is a factor and no higher power of x − r is, r is a zero of multiplicity k. At a zero of odd multiplicity the sign of f changes and the graph crosses the axis; at even multiplicity the sign stays the same and the graph touches the axis and turns back. The higher the multiplicity, the flatter the graph near the zero.
Turning points
A polynomial of degree n has at most n − 1 turning points, where the graph changes from rising to falling or back. A cubic has two turning points or none, never three.
Sketching from factored form
Mark the zeros and their multiplicities, find the y-intercept f(0), and start from the end behavior on the left. Moving right, cross or bounce at each zero. That gives a reliable sketch, though locating the turning points exactly takes calculus.
Common mistakes
- Taking end behavior from the first term as written instead of the term of highest degree.
- Treating a zero of multiplicity 2 as a crossing point: the graph touches the axis and turns back.
- Forgetting a leading coefficient hidden in factored form: −(x + 2)(x − 1)²(x − 3) has leading coefficient −1.
- Expecting n turning points from a polynomial of degree n: there are at most n − 1.
Key terms
- Polynomial
- An expression made by adding terms, each a number times whole-number powers of the variables, such as 3x² − 5x + 2. An expression with a variable in a denominator, under a root or inside a function like sin is not a polynomial.
- Polynomial degree
- The highest power of the variable with a nonzero coefficient, such as 3 for 2x³ − x + 5. With several variables, add the exponents in each term and take the largest sum.
- Leading coefficient
- The coefficient of the highest-power term, such as 2 in 2x³ − x + 5. With the degree, it decides the graph’s end behavior.
- End behavior
- How a function behaves as x tends toward positive or negative infinity. For a polynomial, the degree and leading coefficient determine the eventual directions of both ends.
- Root of an equation
- A value that makes an equation true. A root of f(x) = 0 is also called a zero of f, and it must be in the domain.
- Root multiplicity
- How many times a factor repeats at a root: (x − 2)³ gives the root 2 with multiplicity 3. At an odd multiplicity the graph crosses the x-axis; at an even one it touches and turns back.
- Turning point
- A point where a graph changes from rising to falling or from falling to rising. A polynomial of degree n has at most n − 1 turning points.
- x-intercept
- A point where a graph crosses or touches the x-axis, so its y-value is 0. For y = f(x), its x-value is a zero (root) of f.
Work through an example
Sketch f(x) = −(x + 2)(x − 1)²(x − 3): find its degree, end behavior, zeros with their multiplicities, and y-intercept.
Sketch a polynomial from its zeros →Sources and scope
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Try in the workspace
Open the example inputs, change a value and keep a useful result on your board.
Graph the polynomial Check the expansion in Math Open worked example on a board Degree and leading term in Math ReferenceYour existing work stays on this device. Examples open as editable copies.