BackPolynomials: Vocabulary, Operations, and Multiplication Rules
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Polynomials: Vocabulary and Combining Like Terms
Polynomial Vocabulary
Understanding the terminology associated with polynomials is essential for performing algebraic operations. The following definitions are foundational for working with polynomials:
Monomial: An algebraic expression with only one term (e.g., 7x or 5).
Binomial: An algebraic expression with exactly two terms (e.g., x + 3).
Trinomial: An algebraic expression with exactly three terms (e.g., x^2 + 2x + 1).
Degree of a Monomial: The sum of the exponents of all variables in the monomial. Example: The degree of 3x^2y^3 is 2 + 3 = 5.
Degree of a Polynomial: The greatest degree among all the terms in the polynomial. Example: For 4x^3 + 2x^2 + 7, the degree is 3.
Adding and Subtracting Polynomials
Polynomials can be added or subtracted by combining like terms, which are terms with the same variable(s) raised to the same power(s).
Adding Polynomials: Align like terms vertically and add their coefficients.
Subtracting Polynomials: Distribute the subtraction sign, rewrite as addition of the opposite, and combine like terms.
Example (Addition):
Vertical addition: 3x - 7 + 4x - 2
Example (Subtraction):
Step 1: Rewrite as addition of the opposite: Step 2: Remove parentheses: Step 3: Combine like terms:
Standard form means writing the term with the variable first, followed by the constant.
Multiplying Monomials
Rules for Exponents
When multiplying monomials, exponent rules are used to simplify expressions efficiently.
Product Rule for Exponents: When multiplying like bases, add the exponents. Example:
Power Rule for Exponents: When raising a power to another power, multiply the exponents. Example:
Product to a Power Rule: When raising a product to a power, apply the exponent to each factor inside the parentheses. Example:
Multiplying Polynomials
FOIL Method
The FOIL method is a systematic way to multiply two binomials. FOIL stands for First, Outer, Inner, Last, referring to the pairs of terms to multiply.
First: Multiply the first terms in each binomial.
Outer: Multiply the outer terms.
Inner: Multiply the inner terms.
Last: Multiply the last terms in each binomial.
Example:
First: Outer: Inner: Last: Combine:
Product of the Sum and Difference of Two Terms
Multiplying the sum and difference of the same two terms results in the difference of their squares.
Formula:
Application: This identity is useful for simplifying expressions and factoring.
Example:
