BackMultiplying Polynomials and Exponent Rules (Section R3.2)
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R3.2 Multiplying Polynomials
Exponents
Exponents are used to represent repeated multiplication of the same factor in a compact form. Understanding exponent rules is essential for multiplying polynomials efficiently.
Exponential Form: An expression like means is multiplied by itself three times.
Expanded Form:
Base: The number or variable being multiplied (e.g., in ).
Exponent: The number of times the base is used as a factor (e.g., $3y^3$).
Product Rule for Exponents
The product rule allows us to combine exponents when multiplying like bases.
Rule: For any real number and positive integers and :
Example a:
Example b:
Example c:
Power Rule for Exponents
The power rule is used when raising a power to another power.
Rule: For any real number and positive integers and :
Example a:
Example b:
Example c:
Products to Powers Rule for Exponents
This rule applies when raising a product to a power.
Rule: For real numbers and , and positive integer :
Example a:
Example b:
Example c:
Multiplying Monomials
To multiply monomials (single-term expressions), multiply the coefficients and then apply the product rule to the variables.
Steps:
Multiply the coefficients (numerical parts).
Multiply the variables by adding their exponents.
Example a:
Example b:
Example c:
Multiplying a Monomial and a Polynomial
To multiply a monomial by a polynomial, use the distributive property: multiply the monomial by each term in the polynomial.
Example a:
Example b:
Multiplying Binomials (FOIL Method)
When multiplying two binomials, multiply each term in the first binomial by each term in the second, then combine like terms. The FOIL method (First, Outer, Inner, Last) is a common approach.
FOIL:
First: Multiply the first terms in each binomial.
Outer: Multiply the outer terms.
Inner: Multiply the inner terms.
Last: Multiply the last terms.
Example a:
First:
Outer:
Inner:
Last:
Combine:
Example b:
First:
Outer:
Inner:
Last:
Combine:
Example c:
First:
Outer:
Inner:
Last:
Combine:
