Precalculus Polynomial and Rational Functions: Study Notes and Practice Problems

Polynomial and Rational Functions

Solving Polynomial Equations

Solving polynomial equations involves finding the values of x that satisfy a given equation. This often requires algebraic manipulation, factoring, or applying the quadratic formula for second-degree polynomials.

• Key Point: To solve an equation like $g(3x) - 7 = 9$, isolate the variable and solve for x.

• Example: If $g(3x) = 16$, and $g(x) = x^2$, then $g(3x) = (3x)^2 = 9x^2$, so $9x^2 = 16$ leads to $x = \pm\frac{4}{3}$.

Polynomial Long Division

Polynomial long division is a method for dividing a polynomial by another polynomial of lower or equal degree, similar to numerical long division.

• Key Point: Arrange both polynomials in descending order of degree before dividing.

• Formula: If $f(x)$ is divided by $d(x)$, then $f(x) = d(x) \cdot q(x) + r(x)$, where $q(x)$ is the quotient and $r(x)$ is the remainder.

• Example: Divide $x^4 - 3x^4 + 7x - 5$ by $x^4 + 2$ using long division.

Graphing Rational Functions and Analyzing Behavior

To graph rational functions, determine their end behavior, zeros, and asymptotes. The zeros are found by setting the numerator equal to zero, and vertical asymptotes occur where the denominator is zero (and the numerator is not zero at those points).

• Key Point: End behavior is determined by the degrees of the numerator and denominator.

• Example: For $f(x) = -6(x+3)(x-2)^2(x-4)^2/(x+1)^3$, zeros are at $x = -3, 2, 4$; vertical asymptote at $x = -1$.

Synthetic Division

Synthetic division is a shortcut for dividing a polynomial by a linear factor of the form $(x - c)$.

• Key Point: Only works when dividing by a linear factor.

• Example: Divide $(3x^4 + 12x^3 - x^2 + 10)$ by $(x + 4)$ using synthetic division.

Rational Root Theorem and Listing Rational Zeros

The Rational Root Theorem helps identify possible rational zeros of a polynomial by considering the factors of the constant term and the leading coefficient.

• Key Point: Possible rational zeros are $\pm\frac{p}{q}$, where $p$ divides the constant term and $q$ divides the leading coefficient.

• Example: For $f(x) = -6x^4 - 2x^3 + 5x + 13$, list all possible rational zeros using the Rational Root Theorem.

Logarithmic and Exponential Expressions

Logarithmic and exponential expressions can often be simplified using properties of logarithms and exponents.

• Key Point: $\log_b(MN) = \log_b M + \log_b N$ and $\log_b\left(\frac{M}{N}\right) = \log_b M - \log_b N$.

• Example: Write $\log_5\left(\frac{\sqrt{x+5} \cdot (x-2)^4}{5x^2 (x+1)^3}\right)$ as a sum or difference of logarithms.

Constructing Polynomials from Zeros

Given a set of zeros, a polynomial can be constructed by multiplying factors of the form $(x - r)$ for each zero $r$.

• Key Point: For zeros at $-1, -2, 3$, the polynomial is $f(x) = a(x+1)(x+2)(x-3)$, where $a$ is a leading coefficient.

Solving Logarithmic Equations

To solve logarithmic equations, use properties of logarithms to combine or expand terms, then exponentiate both sides to solve for the variable.

• Key Point: $\log_b(x) = y$ implies $x = b^y$.

• Example: Solve $2\log_2(x) - \log_5(x + 4) = 4$ for $x$.

Asymptotes and Graphing Rational Functions

Asymptotes are lines that the graph of a function approaches but never touches. Rational functions can have vertical, horizontal, or oblique (slant) asymptotes.

• Key Point: Vertical asymptotes occur where the denominator is zero; horizontal asymptotes depend on the degrees of numerator and denominator.

• Example: For $f(x) = \frac{-2x^2 + 12}{3x^2 + 8x + 2}$, find all asymptotes and sketch the graph.

Descartes' Rule of Signs

Descartes' Rule of Signs provides a way to determine the possible number of positive and negative real zeros of a polynomial by counting sign changes in the coefficients.

• Key Point: The number of positive real zeros equals the number of sign changes in $f(x)$ or less by an even number; for negative real zeros, apply the rule to $f(-x)$.

• Example: For $f(x) = 4x^7 + x^6 - 2x^5 + 3x^4 + 9x^2 - 6x^2 - 8x - 10$, count sign changes to determine possible numbers of positive and negative real zeros.

Summary Table: Types of Asymptotes in Rational Functions

Type of Asymptote	How to Find	Example
Vertical	Set denominator equal to zero and solve for x	$x = a$ where denominator is zero
Horizontal	Compare degrees of numerator and denominator	If degrees equal, $y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}$
Oblique (Slant)	If degree of numerator is one more than denominator, divide numerator by denominator	Result is a linear equation $y = mx + b$