M328 Algebra II - Chapter R Part 1

This set covers algebraic properties of real numbers, exponent rules, polynomial definitions/degrees, and the classification of number sets.

Commutative Property for Addition

$a + b = b + a$; terms can switch order.

Commutative Property for Multiplication

$ab = ba$; terms can switch order.

Associative Property for Addition

$(a + b) + c = a + (b + c)$; terms can regroup but must stay in the same order.

Associative Property for Multiplication

$(ab)c = a(bc)$; terms can regroup but must stay in the same order.

Identity for Addition

$a + 0 = a$ and $0 + a = a$; the result is the same.

Identity for Multiplication

$a \times 1 = a$ and $1 \times a = a$; the result is the same.

Inverse for Addition

$a + (-a) = 0$ and $-a + a = 0$; addition of opposites.

Inverse for Multiplication

$a \times \frac{1}{a} = 1$ and $\frac{1}{a} \times a = 1$; multiplication of reciprocals.

Distributive Property of Multiplication over Addition

$a(b + c) = ab + ac$.

Exponent Product Rule

If bases are the same, keep the base and add the exponents: $x^a \times x^b = x^{a+b}$.

Exponent Power of a Power Rule

Keep the base and multiply the exponents: $(x^a)^b = x^{ab}$.

Exponent Quotient Rule

If bases are the same, keep the base and subtract the exponents: $\frac{x^a}{x^b} = x^{a-b}$.

Zero Exponent Rule

Any base (except $0$) to the power of zero is $1$: $x^0 = 1$.

Polynomial

Many terms whose exponents are whole numbers.

Degree of a Polynomial (Single Variable)

The highest exponent in the polynomial.

Degree of a Polynomial (Multiple Variables)

The highest sum of exponents within a single term.

Monomial

A polynomial consisting of $1$ term.

Binomial

A polynomial consisting of $2$ terms.

Trinomial

A polynomial consisting of $3$ terms.

Natural Numbers ($N$)

Also known as counting numbers: $1, 2, 3...$

Whole Numbers ($W$)

Consists of the set of counting numbers and zero: $0, 1, 2, 3...$

Integers ($Z$)

The set of whole numbers and their opposites: $-3, -2, -1, 0, 1, 2, 3...$

Rational Numbers ($Q$)

Numbers you can make by dividing one integer by another, such as $\frac{1}{2}$ or $-1.4$.

Irrational Numbers ($I$)

Any real number that is not rational, such as $\pi$ or $\sqrt{17}$.

Real Numbers ($R$)

Any number anywhere on the number line.

Product of Powers Rule

When multiplying powers with the same base, add the exponents: $x^a \cdot x^b = x^{a+b}$.

Power of a Product Rule

To raise a product to a power, raise each factor to the power: $(ab)^n = a^n b^n$.

Power of a Quotient Rule

To raise a quotient to a power, raise both the numerator and denominator to the power: $(\frac{a}{b})^n = \frac{a^n}{b^n}$.