Polynomials and Exponents

If a and b are real numbers, and m and n are whole non-zero numbers, then

• Product property: a a = a
• Power property: (a) = a
• Product to a power: (abc) = abc
• Quotient property: a/a = a if m>n
• Quotient property: a/a = 1/(a) if n>m
• Quotient to a power property: (a/b) = a/b
• Zero exponent: a = 1
• Properties of - exponents: a = 1/(a) and 1/(a) = a
• Quotient to a - exponent: (a/b) = (b/a)

• Product rule for exponents - when we multiply two numbers or expressions in exponent form with the same base, we keep the base the same and add exponents

a a = a (In examples abc are real numbers, m and n are counting numbers)

• Power to power rule for exponents - when we have a power raised to another power, we keep our base the same and we multiply the exponents

(a) = a

• Power of a product rule for exponents - we can raise a product to a power by raising each factor to the power - useful for variables

(3 x 2) = 3 x 2

(abc) = abc

• Power of a quotient rule for exponents - we can raise a quotient to a power by raising the numerator (dividend) and denominator (divisor) to the power

(a/b) = a/b

3 = 1x3x3x3 = 27 3 = 1x3x3 = 9 3 = 1x3 = 3 3 = 1 0 = undefined 3 = 1/3 3 = 1/9 3 = 1/27

Any nonzero number raised to a negative exponent is not in standard form. We will need to do some rearranging.

Easier: take the reciprocal of the base and make the exponent positive:

3 = 3/1 = 1/3 || 1/9 = (1/9)/1 = 1/9

• Quotient rule - when we divide two numbers or expressions in exponent form and the bases are the same, we can keep the base the same and subtract the exponent in the denominator away from the exponent in the numerator

a/a = a if m>n

a/a = 1/(a) if n>m

a/a = a = a

• Identify polynomials, monomials, binomials, and trinomials
• Determine the degree of polynomials and put in standard form
• Add and subtract monomials
• Add and subtract polynomials
• Evaluate a polynomial for a given value

• Multiply a polynomial by a monomial
• Multiply a binomial by a binomial
• Multiply a trinomial by a binomial

• Square a binomial using the Binomial Squares Pattern
• Multiply conjugates using the Product of Conjugates Pattern
• Recognize and use the appropriate special product pattern

• Simplify expressions using the Quotient Property for Exponents
• Simplify expressions with zero exponents
• Simplify expressions using the quotient to a Power Property
• Simplify expressions by applying several properties
• Divide monomials

• Divide a polynomial by a monomial
• Divide a polynomial by a binomial