College Algebra: Polynomials, Rational Expressions, and Graphing Study Guide

Polynomials

Definition and Standard Form

A polynomial is an algebraic expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. The standard form of a polynomial arranges the terms in descending order of degree.

• Example: $6x^2 - 8x + 8$ is a polynomial in standard form.

• Non-example: $\frac{2x + 5}{x}$ is not a polynomial because it contains a variable in the denominator.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the expression.

• Example: $x^4 + 2x^3 + 7x^2 + 22$ has degree 4.

• Example: $10x^3 - 5x^2 + 4x + 4x^2 - 3$ (combine like terms first) has degree 3.

Operations with Polynomials

Polynomials can be added, subtracted, and multiplied by combining like terms and applying distributive properties.

• Addition/Subtraction: Combine like terms (terms with the same variable and exponent).

• Multiplication: Use distributive property or FOIL for binomials.

• Example: $(2x^3 + 5x^2 - 6x + 9) + (8x^3 + 7x^2 - 7x + 6)$ Combine like terms: $10x^3 + 12x^2 - 13x + 15$

Factoring Polynomials

Factoring is the process of expressing a polynomial as a product of its factors. If a polynomial cannot be factored over the integers, it is called prime.

• Common Methods:

  • Factoring out the greatest common factor (GCF)

  • Factoring trinomials

  • Difference of squares: $a^2 - b^2 = (a - b)(a + b)$

  • Factoring by grouping

• Example: $x^2 - 9 = (x - 3)(x + 3)$

• Prime Example: $x^2 + 7$ is prime over the integers.

Special Products

• Square of a Binomial: $(a + b)^2 = a^2 + 2ab + b^2$

• Product of Sum and Difference: $(a + b)(a - b) = a^2 - b^2$

Rational Expressions

Definition and Domain

A rational expression is a fraction in which the numerator and/or denominator are polynomials. The domain of a rational expression excludes values that make the denominator zero.

• Example: $\frac{x^2 - 9}{x^2 - 25}$ is undefined for $x = 5$ and $x = -5$.

Simplifying Rational Expressions

To simplify, factor both numerator and denominator and cancel common factors. Always state restrictions on the variable.

• Example: $\frac{x^2 - 9}{x^2 - 25} = \frac{(x - 3)(x + 3)}{(x - 5)(x + 5)}$

Operations with Rational Expressions

• Addition/Subtraction: Find a common denominator, combine numerators, and simplify.

• Multiplication/Division: Multiply numerators and denominators, then simplify. For division, multiply by the reciprocal.

• Example (Addition): $\frac{2x}{x^2 - 9} + \frac{3}{x^2 - 9} = \frac{2x + 3}{x^2 - 9}$

• Example (Multiplication): $\frac{x}{x + 2} \times \frac{x + 2}{x - 3} = \frac{x}{x - 3}$ (for $x \neq -2, 3$)

Excluding Values from the Domain

Set the denominator equal to zero and solve for the variable to find excluded values.

• Example: $\frac{3x}{x^2 - 25}$ is undefined for $x = 5$ and $x = -5$.

Graphing and Coordinate Systems

Plotting Points

Points are plotted in the rectangular (Cartesian) coordinate system using ordered pairs $(x, y)$.

• Example: The point $(5, 3)$ is 5 units right and 3 units up from the origin.

Graphing Equations

To graph an equation, plot points that satisfy the equation and connect them smoothly.

• Linear Equations: $y = mx + b$ is a straight line with slope $m$ and y-intercept $b$.

• Quadratic Equations: $y = ax^2 + bx + c$ is a parabola.

• Example: $y = x^2 + 4$ is a parabola opening upward, shifted 4 units up.

Intercepts

• x-intercept: The point(s) where the graph crosses the x-axis ($y = 0$).

• y-intercept: The point where the graph crosses the y-axis ($x = 0$).

• Example: For $y = x - 4$, x-intercept is $4$, y-intercept is $-4$.

Summary Table: Types of Polynomial Factoring

Type	Form	Factoring Method	Example
Greatest Common Factor (GCF)	$ax + ay$	Factor out $a$	$3x + 6 = 3(x + 2)$
Trinomial	$ax^2 + bx + c$	Find two numbers that multiply to $ac$ and add to $b$	$x^2 + 5x + 6 = (x + 2)(x + 3)$
Difference of Squares	$a^2 - b^2$	$(a - b)(a + b)$	$x^2 - 9 = (x - 3)(x + 3)$
Prime	Cannot be factored over integers	State as prime	$x^2 + 7$

Key Formulas and Properties

• Standard Form of a Polynomial: $a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$

• Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Slope of a Line: $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Point-Slope Form: $y - y_1 = m(x - x_1)$

Additional info: This guide covers foundational College Algebra topics including polynomials, rational expressions, factoring, and graphing, as reflected in the provided materials.