College Algebra: Polynomials, Factoring, Equations, and Inequalities Study Guide

Polynomials

Definition and Properties

A polynomial is an algebraic expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents.

• Degree of a Polynomial: The highest power of the variable in the polynomial.

• Leading Coefficient: The coefficient of the term with the highest degree.

• Standard Form: A polynomial written with terms in descending order of degree.

Example: For $2x^4 - 3x^2 + 12x^3 - 2$, the degree is 4 and the leading coefficient is 2.

Identifying Polynomials

• An expression is a polynomial if all exponents of variables are non-negative integers and there are no variables in denominators or under radicals.

• Expressions like $x^4 - x^{-8}$ are not polynomials because of the negative exponent.

Operations with Polynomials

Multiplying Polynomials

To multiply polynomials, use the distributive property (also known as FOIL for binomials) and combine like terms.

• Example: $(x^2 - 7)(x^2 + 7) = x^4 - 49$

Factoring Polynomials

Factoring is the process of writing a polynomial as a product of its factors.

• Common Factoring Methods:

  • Factoring out the Greatest Common Factor (GCF)

  • Factoring by Grouping

  • Factoring Trinomials

  • Factoring the Difference of Squares: $a^2 - b^2 = (a - b)(a + b)$

  • Factoring Completely: Continue factoring until all factors are prime polynomials.

• Prime Polynomial: A polynomial that cannot be factored further over the integers.

Example: $x^2 + 2x - 48 = (x + 8)(x - 6)$

Rational Expressions and Operations

Reducing Rational Expressions

To reduce a rational expression, factor the numerator and denominator and cancel common factors.

• Example: $\frac{x^2 + 2x}{4x}$ simplifies to $\frac{x + 2}{4}$

Adding and Subtracting Rational Expressions

• Find a common denominator, rewrite each fraction, and combine numerators.

• Example: $\frac{2x}{5} - \frac{5x}{3} = \frac{6x - 25x}{15} = \frac{-19x}{15}$

Radicals and Rational Exponents

Simplifying Radical Expressions

• Use properties of radicals: $\sqrt{a^2} = |a|$ and $\sqrt{ab} = \sqrt{a}\sqrt{b}$

• Example: $\sqrt{216x^4} = 6x^2\sqrt{6}$

Operations with Radicals

• Combine like radicals and rationalize denominators if needed.

• Example: $\frac{2\sqrt{35}}{5\sqrt{7}} = \frac{2\sqrt{5}}{5}$

Solving Equations

Linear Equations

• Isolate the variable using inverse operations.

• Example: $2x + 4 = 22 \implies x = 9$

Quadratic Equations

• Can be solved by factoring, completing the square, or using the quadratic formula:

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

• Example: $x^2 - 4x = 0 \implies x(x - 4) = 0 \implies x = 0, 4$

Equations with Radicals

• Isolate the radical and square both sides to eliminate it, then solve the resulting equation.

• Check for extraneous solutions.

Equations with Complex Numbers

• Use $i = \sqrt{-1}$ to express solutions in terms of complex numbers.

• Example: $\sqrt{-9} = 3i$

Inequalities

Solving Linear Inequalities

• Solve as you would an equation, but reverse the inequality sign when multiplying or dividing by a negative number.

• Express solutions in interval notation and graph on a number line.

Example: $2x - 5 > 9 \implies x > 7$

Absolute Value Equations and Inequalities

• For $|ax + b| = c$, write two equations: $ax + b = c$ and $ax + b = -c$.

• For $|ax + b| < c$, the solution is $-c < ax + b < c$.

Complex Numbers

Standard Form

• A complex number is written as $a + bi$, where $a$ is the real part and $b$ is the imaginary part.

• Example: $4 + 6i$

Summary Table: Factoring Methods

Type of Polynomial	Factoring Method	Example
Common Factor	Factor out GCF	$6x^2 + 9x = 3x(2x + 3)$
Trinomial	Split into two binomials	$x^2 + 5x + 6 = (x + 2)(x + 3)$
Difference of Squares	Use $a^2 - b^2 = (a - b)(a + b)$	$x^2 - 9 = (x - 3)(x + 3)$
Grouping	Group terms and factor	$x^3 + 3x^2 + 2x + 6 = (x^2 + 2)(x + 3)$

Additional info:

• Some questions involve graphing solution sets for inequalities, which is a key skill in College Algebra.

• Complex numbers and their operations are included, which are often covered in the latter part of a College Algebra course.

• All examples and explanations are based on standard College Algebra curriculum.