College Algebra Study Guide: Number Theory, Exponents, Polynomials, and Factoring

Topic 1: Number Theory and Basic Operations

Prime and Composite Numbers

Understanding the difference between prime and composite numbers is fundamental in algebra. A prime number has exactly two distinct positive divisors: 1 and itself. A composite number has more than two positive divisors.

• Prime Number: Only divisible by 1 and itself (e.g., 7, 13).

• Composite Number: Has additional divisors (e.g., 12, 18).

• Example: 181 is prime because its only divisors are 1 and 181.

Prime Factorization

Prime factorization is expressing a number as a product of its prime factors.

• Definition: Breaking down a composite number into a product of prime numbers.

• Example: $126 = 2 \times 3^2 \times 7$

Least Common Multiple (LCM)

The LCM of two or more numbers is the smallest number that is a multiple of each.

• Formula: For numbers $a$ and $b$, $\text{LCM}(a, b)$ is the smallest positive integer divisible by both.

• Example: LCM of 27, 48, and 162 is 432.

Fractions: Writing, Reducing, and Operations

Fractions represent parts of a whole and can be manipulated through various operations.

• Writing with a Given Denominator: Convert $\frac{4}{3}$ to a denominator of 16: $\frac{4}{3} = \frac{21.33}{16}$ (rounded).

• Reducing Fractions: Simplify $\frac{30}{35}$ to $\frac{6}{7}$.

• Operations: $10 - (-5) + 2 - (-10) = 27$

Square Roots and Radicals

Square roots and radicals are used to represent fractional exponents.

• Square Root: $\sqrt{64} = 8$

• Simplifying Radicals: $\sqrt{121} = 11$; $\sqrt[3]{8} = 2$

• Product Rule for Radicals: $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$

Topic 2: Exponents, Polynomials, and Simplification

Evaluating Expressions

Algebraic expressions can be evaluated by substituting values for variables and performing arithmetic operations.

• Example: $6 - 20^2 + (-9) = -403$

• Substitution: For $8x - (y^2)$, if $x = 6$ and $y = 8$, $8 \times 6 - 8^2 = 48 - 64 = -16$

Exponent Rules

Exponents follow specific rules for multiplication, division, and powers.

• Product Rule: $a^m \times a^n = a^{m+n}$

• Quotient Rule: $\frac{a^m}{a^n} = a^{m-n}$

• Power Rule: $(a^m)^n = a^{mn}$

• Negative Exponent Rule: $a^{-n} = \frac{1}{a^n}$

• Example: $x^2y^3 \times x^4y^2 = x^{2+4}y^{3+2} = x^6y^5$

Evaluating with Given Values

Substitute the given values into the expression and simplify.

• Example: $2x^2 - 3x + 7$ for $x = -4$ yields $2(-4)^2 - 3(-4) + 7 = 32 + 12 + 7 = 51$

Simplifying Expressions

Combine like terms and apply exponent rules to simplify expressions.

• Example: $\frac{5x^2}{2x^5}$ simplifies to $\frac{5}{2x^3}$

Topic 3: Equations and Solutions

Solving Linear Equations

Linear equations can be solved by isolating the variable using inverse operations.

• Example: $x + 1 = 5$; $x = 4$

• Checking Solutions: Substitute the value back into the equation to verify.

Solving Proportions

Proportions are equations that state two ratios are equal.

• Example: $\frac{3x}{7} = \frac{6}{12}$; cross-multiply to solve for $x$.

Topic 4: Polynomials and Factoring

Degree and Classification of Polynomials

The degree of a polynomial is the highest power of the variable. Polynomials are classified by degree and number of terms.

• Degree: $2x^5 + 4x^4$ has degree 5.

• Binomial: Two terms (e.g., $2x^5 + 4x^4$)

Factoring Polynomials

Factoring involves expressing a polynomial as a product of its factors.

• Greatest Common Factor (GCF): The largest factor shared by all terms.

• Example: GCF of $10m^3$, $40m^7$, $100m^4$ is $10m^3$

• Factoring Trinomials: $x^2 + 2x - 35 = (x + 7)(x - 5)$

• Prime Polynomial: Cannot be factored further over the integers.

Factoring Completely

Continue factoring until all factors are irreducible.

• Example: $x^2 - 81 = (x - 9)(x + 9)$

• Example: $5y^2 + 2y - 3 = (5y - 3)(y + 1)$

Summary Table: Key Algebraic Properties

Property	Rule	Example
Product of Powers	$a^m \times a^n = a^{m+n}$	$x^2 \times x^3 = x^5$
Quotient of Powers	$\frac{a^m}{a^n} = a^{m-n}$	$x^5 / x^2 = x^3$
Power of a Power	$(a^m)^n = a^{mn}$	$(x^2)^3 = x^6$
Negative Exponent	$a^{-n} = \frac{1}{a^n}$	$x^{-2} = \frac{1}{x^2}$
Prime Factorization	Express as product of primes	$126 = 2 \times 3^2 \times 7$
LCM	Smallest common multiple	LCM(27, 48, 162) = 432

Additional info:

• Some questions and answers were inferred from context and standard College Algebra curriculum.

• All variables are assumed to be real numbers unless otherwise specified.