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Algebra 2 Unit 3: Polynomial functions

Polynomials of higher degree have predictable shapes. Their zeros, multiplicities and leading term tell you how the graph looks.

Covers: End behavior, zeros and multiplicity, polynomial division, Remainder and Factor theorems, building polynomials from zeros

Study guide

A polynomial function's end behavior is set by its leading term. Even degree means both ends go the same way (up if the leading coefficient is positive, down if negative). Odd degree means the ends go opposite ways (down on the left and up on the right if the leading coefficient is positive). A degree-n polynomial has at most n real zeros and at most n − 1 turning points.

Zeros are x-values where f(x) = 0, the x-intercepts. A factor (x − r) appearing k times gives a zero of multiplicity k. At odd multiplicity the graph crosses the x-axis; at even multiplicity it touches and turns around. Higher multiplicity flattens the graph near the zero.

Polynomial long division and synthetic division divide one polynomial by another. Synthetic division is a shortcut for dividing by (x − c). The Remainder Theorem says the remainder when dividing f(x) by (x − c) is f(c). The Factor Theorem follows: (x − c) is a factor exactly when f(c) = 0. Once you find one zero, divide it out to reduce the degree and find the rest.

You can build a polynomial from its zeros: zeros 2 and −3 give (x − 2)(x + 3). Use a given point to find the leading coefficient a. Complex and irrational zeros come in conjugate pairs for polynomials with real or rational coefficients.

a³ + b³ = (a + b)(a² − ab + b²) and a³ − b³ = (a − b)(a² + ab + b²). The trinomial factor does not factor further over the real numbers. For a cubic like x³ + 3x² − 4x − 12, group it in pairs, factor each pair, x²(x + 3) − 4(x + 3), then pull out the common binomial to get (x + 3)(x² − 4) = (x + 3)(x − 2)(x + 2). Set each factor to zero to find the zeros.

An expression like x⁴ − 13x² + 36 is quadratic in x², so let u = x² to get u² − 13u + 36 = (u − 4)(u − 9). Substitute back and keep factoring: (x − 2)(x + 2)(x − 3)(x + 3).

Key ideas

• End behavior comes from the leading term.
• Each zero r gives a factor (x − r).
• Odd multiplicity crosses the x-axis; even multiplicity bounces.
• Remainder theorem: dividing by (x − c) leaves remainder f(c).
• Synthetic division is a shortcut for dividing by (x − c).

Formulas and rules

Common mistakes

• Sign errors: zero 3 gives factor (x − 3), not (x + 3).
• Forgetting a zero term when doing synthetic division.

Words to know

End behavior of an odd-degree polynomial with positive leading coefficient

Down on the left, up on the right.

Factor theorem

x = c is a zero exactly when (x − c) is a factor.

Remainder theorem

f(x) ÷ (x − c) has remainder f(c).

Even multiplicity on a graph

The graph touches the axis and turns.

Max number of real zeros of a degree-n polynomial

n

End behavior

The direction of a graph's ends, set by the leading term.

Multiplicity

How many times a factor repeats; odd crosses the axis, even touches.

Synthetic division

A shortcut for dividing a polynomial by (x − c).

Polynomial long division

Dividing polynomials like numbers: divide, multiply, subtract, bring down.

Rational Root Theorem

Possible rational zeros are ±(factors of the constant)/(factors of the leading coefficient).

Turning points

A degree-n polynomial has at most n − 1.

Sample practice questions

25 more questions like these, from level 1 to 5, are in the practice set.