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Study Guide - Smart Notes
Tailored notes based on your materials, expanded with key definitions, examples, and context.
Polynomial Functions
Definition and Structure
A polynomial function of degree n is defined by:
Degree: The highest power of x in the polynomial.
Leading coefficient: The coefficient of the term with the highest degree.
Leading term: The term with the highest degree.
Constant term: The term without x (i.e., ).
Example: For , the degree is 4, the leading coefficient is 6, the leading term is , and the constant term is -2.
Identifying Polynomial Functions
A function is a polynomial if all exponents of x are nonnegative integers and all coefficients are real numbers.
Functions such as or are not polynomials because they involve negative or non-integer exponents, or the variable is in the exponent.
Classification of Polynomials
Polynomials are often classified by their degree and number of terms. Common names include:
Polynomial | Degree | Number of Terms | Name |
|---|---|---|---|
2 | 1 | Quadratic (monomial) | |
2 | 2 | Quadratic (binomial) | |
3 | 3 | Cubic (trinomial) | |
4 | 3 | Quartic (trinomial) |
Operations on Polynomials
Addition and Subtraction
To add or subtract polynomials, combine like terms (terms with the same degree).
Example: ,
Multiplication (Distributive Property)
The distributive property, , is used to multiply polynomials. For binomials, the FOIL method (First, Outer, Inner, Last) is often used.
Example:
Example:
Example:
Division of Polynomials
Polynomials can be divided by monomials or other polynomials. The division algorithm states:
: divisor
: quotient
: remainder (degree less than )
Example: Divide by :
Long Division of Polynomials
To divide polynomials, arrange terms in descending order and divide step by step, subtracting multiples of the divisor.
Example: divided by :
First term:
Multiply and subtract, repeat for next terms.
Factoring Polynomials
Factoring Techniques
Greatest Common Factor (GCF): Factor out the largest common factor from all terms.
Factoring by grouping: Group terms to factor common binomials.
Difference of squares:
Trinomials: Factor as where and (for )
Sum/difference of cubes: ,
Example:
Example:
Example: cannot be factored over the real numbers.
Factoring Strategies
Always look for the GCF first.
Identify the number of terms (binomial, trinomial, etc.).
Check for special forms (difference of squares, cubes).
If the leading coefficient is not 1, use grouping or trial and error.
Check your answer by multiplying all factors.
Practice Problems
Find the sum, difference, product, and quotient of given polynomials.
Factor polynomials using the techniques above.
Write polynomials as products of linear factors when possible.
Summary Table: Factoring Methods
Method | Form | Example |
|---|---|---|
GCF | ||
Grouping | ||
Difference of Squares | ||
Trinomial | ||
Sum/Difference of Cubes |
Additional info: These notes cover polynomial operations, classification, and factoring, which are central topics in College Algebra (Chapter 4: Polynomial Functions).
