BackOperations with Signed Numbers, Polynomials, and Linear Equations: College Algebra Review
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Operations with Signed Numbers and Polynomials
Evaluating Expressions
Evaluating algebraic expressions involves substituting values for variables and performing arithmetic operations according to the order of operations (PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction).
Signed Numbers: Numbers with a positive or negative sign. Operations follow standard arithmetic rules, with attention to sign changes.
Example: Evaluate
Sum:
Scientific and Ordinary Notation
Numbers can be written in scientific notation (as a product of a number between 1 and 10 and a power of 10) or ordinary notation (standard decimal form).
Scientific Notation: , where and is an integer.
Example:
Ordinary to Scientific:
Addition and Subtraction of Polynomials
Polynomials are algebraic expressions consisting of terms with variables raised to whole-number exponents. To add or subtract polynomials, combine like terms (terms with the same variable and exponent).
Example:
Combine terms:
Combine terms:
Result:
Multiplication and Division of Polynomials
Multiplying Polynomials
To multiply polynomials, use the distributive property (also known as FOIL for binomials) and combine like terms.
Example:
First:
Outer:
Inner:
Last:
Combine:
Dividing Polynomials
To divide polynomials, divide each term in the numerator by the denominator, or use long division/synthetic division for more complex expressions.
Example:
Solving Linear Equations and Inequalities
Solving Linear Equations
A linear equation in one variable can be solved by isolating the variable using inverse operations (addition, subtraction, multiplication, division).
Example:
Subtract 4:
Divide by 2:
Solving Linear Inequalities
Linear inequalities are solved similarly to equations, but when multiplying or dividing both sides by a negative number, the inequality sign reverses.
Example:
Add 2:
Divide by 3:
Graphing Solutions
Solutions to inequalities are often represented on a number line, with open or closed circles indicating whether endpoints are included.
Word Problems Involving Linear Equations
Translating Words to Equations
Word problems require translating verbal statements into algebraic equations, then solving for the unknown.
Example: "Five times a number is 25. What is the number?"
Let be the number:
Divide by 5:
Summary Table: Key Operations with Polynomials
Operation | Example | Key Steps |
|---|---|---|
Addition | Combine like terms: , | |
Subtraction | Distribute negative, combine like terms: , | |
Multiplication | FOIL/distribute, combine like terms | |
Division | Divide coefficients and subtract exponents: |
Additional info: These topics are foundational for College Algebra and are essential for understanding more advanced algebraic concepts, including factoring, rational expressions, and functions.
