Topic 2 of 8
Polynomial functions and their roots
Step 13 · Algebra 2 topics
Read a polynomial's end behavior from its degree and leading coefficient, connect its roots to its factors and its graph, and divide polynomials, using the remainder theorem.
Degree and end behavior
A polynomial function's degree is its highest power. The degree and the sign of the leading coefficient decide what the graph does at the far left and the far right.
| Degree | Leading coefficient | Left end | Right end |
|---|---|---|---|
| even | positive | up | up |
| even | negative | down | down |
| odd | positive | down | up |
| odd | negative | up | down |
Example: End behavior
Describe the ends of f(x) = −2x⁴ + x³ − 5.
- The degree, 4, is even, and the leading coefficient, −2, is negative.
AnswerBoth ends go down
Roots and factors
If r is a root, so that f(r) = 0, then x − r is a factor, and the other way round. The graph meets the x-axis at each real root.
Example: From factors to roots
What are the roots of f(x) = (x − 1)(x + 3)²?
- x − 1 = 0 gives 1.
- (x + 3)² = 0 gives −3, a double root.
Answer1, and −3 twice
Example: From roots to a polynomial
Write a polynomial with roots 2, −1 and 4.
- Each root r gives a factor x − r.
Answerf(x) = (x − 2)(x + 1)(x − 4)
The fundamental theorem of algebra says that a polynomial of degree n has exactly n roots, counting repeated roots and complex ones.
Dividing polynomials
Divide polynomials by long division, or by synthetic division when dividing by x − r. The remainder theorem says the remainder when f(x) is divided by x − r is f(r).
| r = 1 | x³ | x² | x | 1 |
|---|---|---|---|---|
| coefficients | 1 | −6 | 11 | −6 |
| add r × result | 1 | −5 | 6 | |
| result | 1 | −5 | 6 | 0 |
Example: Synthetic division
Divide x³ − 6x² + 11x − 6 by x − 1.
- Bring down the 1. Multiply by r = 1 and add each time.
- 1 × 1 = 1, and −6 + 1 = −5. −5 × 1 = −5, and 11 − 5 = 6. 6 × 1 = 6, and −6 + 6 = 0.
- The quotient is x² − 5x + 6, with remainder 0.
Answerx² − 5x + 6
Example: Finishing the factoring
Factor x³ − 6x² + 11x − 6 completely.
- From the division: (x − 1)(x² − 5x + 6).
- x² − 5x + 6 = (x − 2)(x − 3).
Answer(x − 1)(x − 2)(x − 3)
Example: The remainder theorem
Find the remainder when f(x) = x³ + 2x − 5 is divided by x − 2.
- f(2) = 8 + 4 − 5 = 7.
Answer7
Try it yourself
Work each one on paper first, then open the answer to check.
Describe the ends of f(x) = 3x⁵ − x².
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Down on the left, up on the right.
At most how many turning points can a degree-4 polynomial have?
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3
What are the roots of (x + 2)(x − 5)²?
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−2, and 5 twice
Write a polynomial with roots 0, 3 and −3.
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x(x − 3)(x + 3), which is x³ − 9x
Divide x² + 5x + 6 by x + 2.
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x + 3
What is the remainder when x³ − 4x + 1 is divided by x − 3?
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16
Is x − 2 a factor of x³ − 8?
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Yes: 2³ − 8 = 0.