BackPrecalculus Study Notes: Exponents and Polynomials
Study Guide - Smart Notes
Tailored notes based on your materials, expanded with key definitions, examples, and context.
Exponents
Definition and Properties of Exponents
Exponents are a shorthand notation for repeated multiplication of the same number or variable. Understanding the rules of exponents is fundamental in algebra and precalculus.
Exponent Notation: means multiplying a by itself n times.
Example:
Product Rule for Exponents
The product rule allows you to combine exponents when multiplying like bases.
Rule:
Example:
Power Rule for Exponents
The power rule is used when raising a power to another power.
Rule:
Example:
Quotient Rule for Exponents
The quotient rule is used when dividing like bases with exponents.
Rule: , where
Example:
Zero Exponent
Any nonzero number raised to the zero power is 1.
Rule: , where
Example:
Negative Exponents
Negative exponents represent reciprocals.
Rule: , where
Example:
Fractional Exponents
Fractional exponents represent roots.
Rule:
Example:
Polynomials
Definition and Degree
A polynomial is an expression consisting of variables and coefficients, involving only non-negative integer exponents.
General Form:
Degree: The highest exponent of the variable in the polynomial.
Example: has degree 4.
Addition and Subtraction of Polynomials
Combine like terms by adding or subtracting their coefficients.
Example:
Multiplication of Polynomials
Multiply each term in the first polynomial by each term in the second, then combine like terms.
Example:
Multiplying Two Binomials (FOIL Method)
FOIL stands for First, Outer, Inner, Last, referring to the terms multiplied together.
Example:
Factoring Polynomials
Factoring is the process of writing a polynomial as a product of its factors.
Greatest Common Factor (GCF): Factor out the largest common factor from all terms.
Example:
Factoring by Grouping: Group terms to factor common binomials.
Example:
Special Factoring Formulas
Difference of Squares:
Sum of Cubes:
Difference of Cubes:
Practice Problems (Selected Examples)
Exponent Practice:
Negative Exponent:
Polynomial Addition:
Polynomial Multiplication:
Factoring:
Summary Table: Exponent Rules
Rule | Formula | Example |
|---|---|---|
Product | ||
Quotient | ||
Power | ||
Zero Exponent | ||
Negative Exponent | ||
Fractional Exponent |
Summary Table: Special Factoring Formulas
Name | Formula |
|---|---|
Difference of Squares | |
Sum of Cubes | |
Difference of Cubes |
Additional info: Some explanations and examples have been expanded for clarity and completeness, based on standard precalculus curriculum.
