Precalculus Polynomial and Rational Functions Syllabus

Polynomial and Rational Functions syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Polynomial and Rational Functions in Precalculus, not a summary of it.

1. 1

   Quadratic Functions and Models

   3 topics

   • Standard and Vertex Forms
   • Graphing Parabolas
   • Modeling with Quadratics
2. 2

   Higher-Degree Polynomial Functions

   4 topics

   • End Behavior and Leading Coefficient Test
   • Zeros and Multiplicity
   • Intermediate Value Theorem
   • Sketching Polynomial Graphs
3. 3

   Polynomial Division

   3 topics

   • Long Division of Polynomials
   • Synthetic Division
   • Remainder and Factor Theorems
4. 4

   Zeros of Polynomial Functions

   4 topics

   • Rational Zero Theorem
   • Fundamental Theorem of Algebra
   • Descartes's Rule of Signs
   • Conjugate Pairs and Complex Zeros
5. 5

   Complex Numbers

   2 topics

   • Imaginary Unit and Standard Form
   • Operations with Complex Numbers
6. 6

   Rational Functions

   4 topics

   • Domain and Discontinuities
   • Vertical and Horizontal Asymptotes
   • Holes and Removable Discontinuities
   • Sketching Rational Functions
7. 7

   Inequalities

   3 topics

   • Polynomial Inequalities
   • Rational Inequalities
   • Sign Charts and Test Intervals

Polynomial and Rational Functions flashcards for Precalculus

22 of 50 cards from the Polynomial and Rational Functions deck — real questions with worked answers.

1. What is the standard form of a quadratic function?

   $f(x) = ax^{2} + bx + c$, where $a \neq 0$.

2. What is the vertex form of a quadratic function, and what does each part represent?

   $f(x) = a(x - h)^{2} + k$, where $(h, k)$ is the vertex and $a$ controls the direction and width of the parabola.

3. For a parabola in standard form $f(x) = ax^{2} + bx + c$, what is the $x$-coordinate of the vertex?

   $x = -\dfrac{b}{2a}$. The $y$-coordinate is found by evaluating $f\!\left(-\dfrac{b}{2a}\right)$.

4. How does the sign of the leading coefficient $a$ affect a parabola's opening?

   If $a > 0$ the parabola opens upward (has a minimum); if $a < 0$ it opens downward (has a maximum).

5. What is the equation of the axis of symmetry of a parabola $f(x) = a(x-h)^{2} + k$?

   The vertical line $x = h$.

6. For a quadratic model $f(x) = ax^{2} + bx + c$ with $a < 0$, how do you find the maximum value?

   The maximum occurs at $x = -\dfrac{b}{2a}$, and the maximum value is $f\!\left(-\dfrac{b}{2a}\right)$ (the $k$-value of the vertex).

7. How does the value of $|a|$ affect the width of a parabola?

   Larger $|a|$ makes the parabola narrower (stretched vertically); smaller $|a|$ (between 0 and 1) makes it wider (compressed).

8. State the Leading Coefficient Test for the end behavior of a polynomial of even degree.

   If the degree is even: with $a_{n} > 0$ both ends rise ($x \to \pm\infty,\ f(x) \to +\infty$); with $a_{n} < 0$ both ends fall ($f(x) \to -\infty$).

9. State the Leading Coefficient Test for the end behavior of a polynomial of odd degree.

   If the degree is odd: with $a_{n} > 0$ the graph falls left, rises right ($x\to-\infty, f\to-\infty$; $x\to+\infty, f\to+\infty$); with $a_{n} < 0$ it rises left, falls right.

10. What is the maximum number of turning points of a polynomial of degree $n$?

    At most $n - 1$ turning points.

11. How does the multiplicity of a real zero affect the graph's behavior at that $x$-intercept?

    Odd multiplicity: the graph crosses the $x$-axis. Even multiplicity: the graph touches the $x$-axis and turns around (does not cross).

12. What does it mean for a zero $c$ of a polynomial to have multiplicity $k$?

    The factor $(x - c)$ appears exactly $k$ times in the factored form, i.e. $(x - c)^{k}$ divides the polynomial.

13. State the Intermediate Value Theorem as applied to locating real zeros of a polynomial.

    If $f$ is continuous (all polynomials are) and $f(a)$ and $f(b)$ have opposite signs, then there is at least one $c$ between $a$ and $b$ with $f(c) = 0$.

14. When sketching a polynomial graph, how do you determine the $y$-intercept?

    Evaluate $f(0)$, which equals the constant term $a_{0}$.

15. In the polynomial long division of $f(x)$ by $d(x)$, what is the Division Algorithm identity?

    $f(x) = d(x)\,q(x) + r(x)$, where $r(x) = 0$ or $\deg r < \deg d$.

16. When can synthetic division be used instead of long division?

    Only when dividing by a linear factor of the form $x - c$ (a first-degree divisor with leading coefficient 1).

17. Using synthetic division to divide by $x - c$, what value do you place in the division box?

    The value $c$ (the zero of the divisor $x - c$).

18. State the Remainder Theorem.

    If a polynomial $f(x)$ is divided by $x - c$, the remainder equals $f(c)$.

19. State the Factor Theorem.

    $x - c$ is a factor of polynomial $f(x)$ if and only if $f(c) = 0$.

20. State the Rational Zero Theorem.

    If $f(x) = a_{n}x^{n} + \cdots + a_{0}$ has integer coefficients, every rational zero has the form $\dfrac{p}{q}$, where $p$ divides the constant term $a_{0}$ and $q$ divides the leading coefficient $a_{n}$.

21. For $f(x) = 2x^{3} + x - 6$, list the possible rational zeros from the Rational Zero Theorem.

    $p$ divides 6: $\pm1, \pm2, \pm3, \pm6$; $q$ divides 2: $\pm1, \pm2$. Possible zeros: $\pm1, \pm2, \pm3, \pm6, \pm\tfrac{1}{2}, \pm\tfrac{3}{2}$.

22. State the Fundamental Theorem of Algebra.

    Every polynomial of degree $n \geq 1$ with complex coefficients has at least one complex zero.

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