AP Precalculus Unit 1A - Polynomial and Rational Functions

Unit 1A Review

Last Update on June 5, 2025

4 mins

This review covers key concepts of polynomial and rational functions, including identifying zeros, analyzing end behaviors, and understanding symmetry.

Unit 1A Review: Polynomial and Rational Functions

This review covers key concepts of polynomial and rational functions, including identifying zeros, analyzing end behaviors, and understanding symmetry.

Examples

Linear Polynomial: f(x) = 2x + 3. Zero at x = -3/2.
Quadratic Polynomial: f(x) = x^2 - 4. Zeros at x = -2, 2.
Cubic Polynomial: f(x) = x^3 - 27. Zero at x = 3.
Quartic Polynomial: f(x) = x^4 - 16. Zeros at x = ±2, ±2i.
Quintic Polynomial: f(x) = x^5 - 32x. Zeros at x = 0, ±2√2, ±i2√2.
Rational Function: f(x) = (x^2 - 9)/(x + 3). Hole at x = -3.
Even Polynomial: f(x) = x^4 + 4x^2. Symmetric about the y-axis.
Odd Polynomial: f(x) = x^3 - x. Symmetric about the origin.
End Behavior: f(x) = 3x^5. As x → ±∞, f(x) → ±∞.
Complex Zeros: f(x) = x^2 + 1. Zeros at x = ±i.
Polynomial Division: Divide x^3 + 2x^2 - 5x - 6 by x - 1.
Synthetic Division: Use synthetic division to divide x^3 - 4x^2 + 5x - 2 by x - 2.
Factoring Polynomials: Factor x^3 - 6x^2 + 11x - 6.
Polynomial Graphs: Sketch the graph of f(x) = (x - 1)^2(x + 2).
Rational Function Asymptotes: Identify vertical and horizontal asymptotes of f(x) = (2x^2 - 8)/(x^2 - 4).
Transformations: Describe transformations of f(x) = x^2 to g(x) = -2(x + 3)^2 + 1.
Descartes' Rule of Signs: Use Descartes' rule on f(x) = x^4 - 3x^3 + 2x^2 + x - 5.
Intermediate Value Theorem: Apply the Intermediate Value Theorem to f(x) = x^3 - x - 1 on the interval [0, 2].
Remainder Theorem: Use the Remainder Theorem to evaluate f(2) for f(x) = x^

Review

appc_unit_1a_review.pdf

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Review Solutions

appc_unit_1a_review_solutions.pdf

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Corrective Assignment

APPC Unit 1A CA.pdf

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`Frequently Asked Questions: Polynomials & Rational Functions`


Is a polynomial function a rational function?
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Yes. A polynomial function P(x) is a special type of rational function. A rational function is defined as the ratio of two polynomial functions, P(x)/Q(x), where Q(x) is not the zero polynomial. Any polynomial P(x) can be written as P(x)/1, and since 1 is a non-zero polynomial (the constant polynomial), all polynomial functions are indeed a subset of rational functions.


Are all rational functions polynomial functions?
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No. While all polynomials are rational functions, not all rational functions are polynomials. A rational function P(x)/Q(x) is only a polynomial if the denominator Q(x) is a non-zero constant (or the division simplifies to a polynomial with no remainder and Q(x) is not zero). For example, f(x) = 1/x or g(x) = (x+1)/(x-1) are rational functions but are not polynomial functions.


What is the difference between a polynomial function and a rational function?
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The key difference lies in their definition:A polynomial function is a sum of terms, each consisting of a constant coefficient multiplied by a variable raised to a non-negative integer power (e.g., ax² + bx + c).
A rational function is the ratio of two polynomial functions, P(x)/Q(x), where Q(x) is not the zero polynomial.
Polynomials are a more restrictive class of functions; they are essentially rational functions where the denominator is a constant. Rational functions can have variables in the denominator, leading to potential asymptotes and domain restrictions where the denominator is zero.


How can you tell if a function is a polynomial or a rational function?
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Examine the function's structure:If the function is written solely as a sum or difference of terms with constant coefficients and non-negative integer powers of the variable (e.g., f(x) = 3x⁴ - 2x + 5), it is a polynomial function.
If the function is written as a fraction where both the numerator and the denominator are polynomial functions, and the denominator is not identically zero (e.g., g(x) = (x² + 1) / (x - 2)), it is a rational function.
Remember, if it fits the polynomial definition, it *also* fits the rational function definition, but rational functions include cases where the denominator has variables.


How do you find the potential rational zeros of a polynomial function?
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You use the Rational Root Theorem. For a polynomial with integer coefficients, any potential rational zero (root) must be of the form p/q, where:'p' is an integer factor of the constant term (the term without a variable).
'q' is an integer factor of the leading coefficient (the coefficient of the term with the highest power).
You list all possible factors for 'p' and 'q' and form all possible fractions ±p/q. These are your potential rational zeros.


How do you find the actual rational zeros of a polynomial function?
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Once you have the list of potential rational zeros from the Rational Root Theorem:Test each potential zero using synthetic division or direct substitution into the polynomial function.
If P(c) = 0 for a potential zero 'c', then 'c' is an actual rational zero of the polynomial, and (x - c) is a factor.
If you find a zero, you can use the result of the synthetic division (the depressed polynomial) to continue searching for other zeros. This process is repeated until the depressed polynomial is quadratic or simple enough to solve by other means (factoring, quadratic formula).


How do you write a polynomial function with rational coefficients given its zeros?
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If 'c' is a zero of a polynomial, then (x - c) is a factor. For a polynomial with rational coefficients, you also need to consider the Conjugate Root Theorem:If (a + bi) is a zero (where b ≠ 0), then its complex conjugate (a - bi) must also be a zero.
If (a + √b) is a zero (where √b is irrational), then its irrational conjugate (a - √b) must also be a zero.
Once you have all the required conjugate pairs:Write the factors corresponding to each zero (x - c).
Multiply these factors together. For conjugate pairs, multiplying their factors [ (x - (a+bi)) * (x - (a-bi)) ] or [ (x - (a+√b)) * (x - (a-√b)) ] will result in a quadratic with rational coefficients.
The resulting polynomial will have rational coefficients. You can multiply by a constant if necessary to meet additional conditions (like a specific point the function must pass through).

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AP Precalculus