Polynomial and Rational Functions: Study Notes and Worked Examples

Polynomial and Rational Functions

Quadratic Functions: Vertex Form and Properties

Quadratic functions are polynomials of degree 2 and can be written in vertex form as $f(x) = a(x-h)^2 + k$. The vertex form is useful for identifying the vertex, axis of symmetry, and the direction in which the parabola opens.

• Axis of Symmetry: The axis of symmetry is the vertical line $x = h$.

• Vertex: The vertex is the point $(h, k)$.

• Domain: For any quadratic function, the domain is $(- , )$.

• Range: If $a < 0$, the range is $(- , k]$; if $a > 0$, the range is $[k, )$.

• Example: For $f(x) = 1 - (x-3)^2$, the vertex is $(3, 1)$, axis of symmetry is $x=3$, domain is $(- , )$, and range is $(- , 1]$.

Identifying Polynomials and Their Degree

A polynomial is an expression consisting of variables and coefficients, involving only non-negative integer powers of the variable. The degree of a polynomial is the highest power of the variable.

• Definition: A polynomial in $x$ is of the form $a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$ where $n$ is a non-negative integer.

• Degree: The degree is $n$, the highest exponent of $x$.

• Non-Polynomial Example: $h(x) = 4x^2 + 2x^2 + 3x^1$ is not a polynomial because $x^1$ is not a positive integer power (should be $x^1$ is fine, but context suggests a typo; possibly $x^{-1}$).

End Behavior of Polynomial Functions

The end behavior of a polynomial function describes how the function behaves as $x$ approaches $+$ or $-$. It is determined by the degree and the leading coefficient.

• Even Degree, Positive Leading Coefficient: Rises to the left and right.

• Even Degree, Negative Leading Coefficient: Falls to the left and right.

• Odd Degree, Positive Leading Coefficient: Falls to the left, rises to the right.

• Odd Degree, Negative Leading Coefficient: Rises to the left, falls to the right.

• Example: For $f(x) = 4x^4 + 3x^2 - x + 5$, degree is 4 (even), leading coefficient is 4 (positive), so the graph rises to the left and right.

Zeros of Polynomial Functions and Multiplicity

The zeros (roots) of a polynomial are the values of $x$ for which $f(x) = 0$. The multiplicity of a zero refers to how many times a particular root occurs.

• Multiplicity: If $(x-a)^k$ is a factor, $x=a$ is a zero of multiplicity $k$.

• Odd Multiplicity: The graph crosses the $x$-axis at $x=a$.

• Even Multiplicity: The graph touches the $x$-axis and turns around at $x=a$.

• Example: For $f(x) = (x-5)(x+3)^2$, $x=5$ is a zero of multiplicity 1 (crosses), $x=-3$ is a zero of multiplicity 2 (touches and turns).

Intermediate Value Theorem

The Intermediate Value Theorem states that if a continuous function $f(x)$ takes on values of opposite sign at two points, then it must cross zero somewhere between those points.

• Application: If $f(a)$ and $f(b)$ have opposite signs, there is at least one real zero between $a$ and $b$.

• Example: For $f(x) = 4x^3 - 2x - 4$, $f(1) = -2$, $f(4) = 244$; since the signs are opposite, there is a real zero between $x=1$ and $x=4$.

Finding Zeros: Factoring and Synthetic Division

To find the zeros of a polynomial, factor the polynomial or use synthetic division.

• Factoring: Express the polynomial as a product of factors and set each factor equal to zero.

• Synthetic Division: A shortcut method for dividing a polynomial by a linear factor $(x-c)$.

• Example: $x^3 + 2x^2 - 5x - 6$ divided by $(x-2)$ yields quotient $x^2 + 4x + 3$ and remainder 0. Factoring gives $(x-2)(x+3)(x+1)$, so zeros are $x=2, -3, -1$.

Possible Rational Zeros: Rational Root Theorem

The Rational Root Theorem provides a way to list all possible rational zeros of a polynomial with integer coefficients.

• Possible Rational Zeros: $rac{ ext{factor of constant term}}{ ext{factor of leading coefficient}}$

• Example: For $a_0 = 1, 2, 4, 8$, possible rational zeros are $\{ , , ,  \}$.

Graphing Polynomial Functions

Graphing involves plotting zeros, identifying multiplicity, determining end behavior, and plotting additional points to confirm the shape.

• Y-intercept: Set $x=0$ and solve for $f(0)$.

• Symmetry: Test for y-axis symmetry ($f(-x) = f(x)$), origin symmetry ($f(-x) = -f(x)$), or neither.

• Example: For $f(x) = -x^4 + x^2$, the function is even and symmetric with respect to the y-axis.

Polynomial Long Division

Polynomial long division is used to divide one polynomial by another, resulting in a quotient and a remainder.

• Process: Divide the highest degree term, multiply, subtract, repeat until degree of remainder is less than divisor.

• Example: $\frac{6x^3 - 8x^2 + x + 1}{6x^2 + 4x + 7}$ yields quotient $x-2$ and remainder $2x+15$.

• Final Answer: $\frac{6x^3 - 8x^2 + x + 1}{6x^2 + 4x + 7} = (x-2) + \frac{2x+15}{6x^2 + 4x + 7}$

Summary Table: End Behavior of Polynomial Functions

Degree	Leading Coefficient	End Behavior
Even	Positive	Rises left and right
Even	Negative	Falls left and right
Odd	Positive	Falls left, rises right
Odd	Negative	Rises left, falls right

Additional info:

• Some explanations and terminology have been expanded for clarity and completeness.

• All equations are provided in LaTeX format for mathematical accuracy.