BackCollege Algebra: Essential Concepts, Operations, and Problem Solving
Study Guide - Smart Notes
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Topic 1: Fundamental Number Concepts and Operations
Prime and Composite Numbers
Understanding the difference between prime and composite numbers is foundational in algebra. A prime number has exactly two distinct positive divisors: 1 and itself. A composite number has more than two positive divisors.
Example: 181 is a prime number because its only divisors are 1 and 181.
Prime Factorization
Prime factorization is expressing a number as a product of its prime factors.
Example:
Least Common Multiple (LCM)
The LCM of two or more numbers is the smallest number that is a multiple of each.
Example: Find the LCM of 27, 48, and 162.
Fractions: Writing and Reducing
Fractions can be rewritten with different denominators and reduced to simplest form.
Example: Write with a denominator of 16: (but typically, use integer values).
Reducing:
Operations with Integers and Fractions
Performing operations and expressing results in lowest terms is essential.
Example:
Example:
Evaluating Expressions and Square Roots
Evaluating algebraic expressions and square roots is a key skill.
Example:
Square root:
Decimal approximation:
Simplifying Radicals
Radicals can be simplified by factoring out perfect squares.
Example:
Product Rule for Radicals
The product rule allows you to multiply radicals: .
Example:
Topic 2: Algebraic Expressions and Exponents
Evaluating Algebraic Expressions
Substitute given values for variables and simplify.
Example:
Example:
Simplifying Expressions with Exponents
Apply exponent rules to simplify expressions.
Product Rule:
Power Rule:
Negative Exponent Rule:
Example:
Multiplying Monomials
Multiply coefficients and add exponents for like bases.
Example:
Using Laws of Exponents
Apply laws to simplify expressions, ensuring all exponents are positive.
Example:
Example:
Topic 3: Equations and Solutions
Solving Linear Equations
Linear equations can be solved by isolating the variable.
Example:
Example:
Checking Solutions
Substitute the solution back into the original equation to verify correctness.
Example: For , if , check by substituting .
Solving Proportions
Proportions are equations that state two ratios are equal.
Example:
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.
Example: has degree 5 and is a binomial.
Factoring Polynomials
Factoring involves expressing a polynomial as a product of its factors.
Greatest Common Factor (GCF): The largest factor that divides each term.
Example: GCF of is .
Factoring Trinomials:
Prime Trinomials: If a trinomial cannot be factored, it is called prime.
Factoring Completely
Continue factoring until all factors are irreducible.
Example:
Example:
Additional info:
Some answers and solutions are provided in the images, reinforcing the concepts.
Topics covered are highly relevant to College Algebra, including number theory, operations, exponents, equations, and polynomials.
