College Algebra: Study Guide for Polynomial, Rational, Radical Functions, and Circles

Module 2 Topic 1: Polynomial Functions & Inequalities

Introduction

Polynomial functions are fundamental in algebra, involving expressions with variables raised to whole number powers. Understanding their structure, properties, and solutions is essential for analyzing more complex mathematical models.

• Polynomial Expressions: A polynomial is an expression of the form $a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$, where the coefficients $a_i$ are real numbers and $n$ is a non-negative integer.

• Terms: Each part of a polynomial separated by a plus or minus sign is called a term. The degree is the highest power of $x$ present.

• Types of Polynomials:

  • Monomial: One term (e.g., $5x^3$)

  • Binomial: Two terms (e.g., $x^2 - 4$)

  • Trinomial: Three terms (e.g., $x^2 + 3x + 2$)

• Identifying Polynomials: An expression is a polynomial if all exponents of the variable are non-negative integers and coefficients are real numbers.

• Leading Coefficient and Degree: The leading coefficient is the coefficient of the term with the highest degree.

• Graph Characteristics: Polynomial graphs can be smooth and continuous, with end behavior determined by the leading term.

• Zeros and Solutions: The zeros of a polynomial are the values of $x$ for which $P(x) = 0$.

• Factoring: Factoring polynomials helps find their zeros and solve equations.

• Leading Coefficient Test: Determines end behavior of the graph based on the sign and degree of the leading term.

• x-intercepts and y-intercept: The x-intercepts are the zeros; the y-intercept is $P(0)$.

• Graphing: Sketching involves plotting intercepts, analyzing end behavior, and identifying turning points.

• Polynomial Inequalities: Solving involves finding intervals where the polynomial is positive or negative.

Example:

Find the zeros of $f(x) = x^2 - 5x + 6$ by factoring: $x^2 - 5x + 6 = (x-2)(x-3) = 0 \implies x = 2, 3$.

Module 2 Topic 2: Polynomial Division & Division Algorithm

Introduction

Polynomial division is a method for dividing one polynomial by another, similar to long division with numbers. The Division Algorithm and Remainder Theorem are key tools in this process.

• Division Algorithm: For polynomials $f(x)$ and $d(x) \neq 0$, there exist unique polynomials $q(x)$ and $r(x)$ such that $f(x) = d(x)q(x) + r(x)$, where the degree of $r(x)$ is less than the degree of $d(x)$.

• Remainder Theorem: The remainder when $f(x)$ is divided by $(x - c)$ is $f(c)$.

• Polynomial Long Division: A step-by-step process similar to numerical long division.

• Synthetic Division: A shortcut for dividing by linear factors of the form $(x - c)$.

• Constructing Polynomials: Given zeros and a point, construct a polynomial formula.

Example:

Divide $f(x) = 2x^3 + 3x^2 - x + 5$ by $x - 2$ using synthetic division.

Module 2 Topic 3: Real Zeros & Graphs of Polynomial Functions

Introduction

Finding the real zeros of polynomials and understanding their graphs are crucial for analyzing function behavior and solving equations.

• Rational Zero Theorem: Provides a list of possible rational zeros for a polynomial with integer coefficients.

• Finding Real Zeros: Use the Rational Zero Theorem and division to test possible zeros.

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

• Fundamental Theorem of Algebra: A polynomial of degree $n$ has exactly $n$ complex zeros (counting multiplicity).

• Turning Points: The graph of a degree $n$ polynomial has at most $n-1$ turning points.

• Behavior at x-intercepts: The graph crosses or touches the x-axis depending on the multiplicity of the zero.

• Graphical Features: Degree, intercepts, multiplicity, and turning points help sketch the graph.

Example:

For $f(x) = x^3 - 4x^2 + x + 6$, possible rational zeros are $\pm1, \pm2, \pm3, \pm6$. Test each to find actual zeros.

Module 2 Topic 4: Rational Functions & Inequalities

Introduction

Rational functions are quotients of polynomials. Their graphs and properties depend on the degrees and factors of the numerator and denominator.

• Rational Function: A function of the form $f(x) = \frac{P(x)}{Q(x)}$, where $Q(x) \neq 0$.

• Key Features: Domain, range, asymptotes (vertical and horizontal), and intercepts.

• Vertical Asymptotes: Occur where $Q(x) = 0$ and $P(x) \neq 0$.

• Holes: Occur where both $P(x)$ and $Q(x)$ have a common factor.

• Horizontal Asymptotes: Determined by comparing degrees of numerator and denominator.

• Graphing: Identify all key features to sketch the graph.

• Rational Inequalities: Solve by finding intervals where the rational expression is positive or negative.

Example:

For $f(x) = \frac{x^2 - 1}{x - 1}$, there is a hole at $x = 1$ and a slant asymptote since the degree of the numerator is one more than the denominator.

Module 2 Topic 5: Rational Equations

Introduction

Rational equations involve rational expressions set equal to each other. Solving them requires finding a common denominator and checking for extraneous solutions.

• Cross-Multiplication: Used when two rational expressions are set equal.

• Least Common Denominator (LCD): Multiply both sides by the LCD to clear denominators.

• Extraneous Roots: Solutions that do not satisfy the original equation due to domain restrictions.

Example:

Solve $\frac{2}{x} = \frac{3}{x+1}$ by cross-multiplying: $2(x+1) = 3x \implies 2x + 2 = 3x \implies x = 2$.

Module 2 Topic 6: Direct & Inverse Variation

Introduction

Direct and inverse variation describe relationships where one variable changes in proportion to another, either directly or inversely.

• Direct Variation: $y = kx$, where $k$ is the constant of variation.

• Inverse Variation: $y = \frac{k}{x}$, where $k$ is the constant of variation.

• Verbal Descriptions: Translate word problems into variation equations.

• Problem Solving: Use the equations to solve for unknowns given values.

Example:

If $y$ varies directly as $x$ and $y = 10$ when $x = 2$, then $k = 5$ and $y = 5x$.

Module 2 Topic 7: Radical Equations

Introduction

Radical equations contain variables within a radical, usually a square root. Solving them often involves isolating the radical and squaring both sides.

• Solving Radical Equations: Isolate the radical, then raise both sides to the appropriate power to eliminate it.

• Extraneous Roots: Always check solutions in the original equation, as squaring can introduce invalid solutions.

Example:

Solve $\sqrt{x+3} = x - 1$. Square both sides: $x + 3 = (x - 1)^2$.

Module 2 Topic 8: Circles

Introduction

The equation of a circle describes all points in a plane at a fixed distance (radius) from a fixed point (center).

• Standard Form: $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius.

• Finding Center and Radius: Identify from the standard form or by completing the square from the general form.

• General Form: $x^2 + y^2 + Dx + Ey + F = 0$.

• Equation from Center and Radius: Substitute values into the standard form.

• Equation from Diameter Endpoints: Find the midpoint (center) and distance (radius).

• Completing the Square: Used to convert the general form to standard form.

Example:

Write the equation of a circle with center $(2, -3)$ and radius $5$:$(x - 2)^2 + (y + 3)^2 = 25$.