**Key Terms**

- Power – A small raised number that tells you how many times to multiply the base number by itself.
- It is also called an exponent.
- Ex.
- Degree is the exponent value
- Highest degree is the highest value of all of the exponents in the polynomial.

- Common Factors – Numbers or expressions that are factors of two or more numbers or polynomials.
- Polynomial – An algebraic expression with one or more terms.
- None of the variables are in the denominator of a fraction, and any exponents are whole numbers.
- Ex. is a polynomial.
- Ex. is a NOT a polynomial because there is a variable (x) in the denominator of the fraction.

- None of the variables are in the denominator of a fraction, and any exponents are whole numbers.

**Review**

- Nomial – Term
- Terms are separated by plus and minus signs.
- Monomial – 1 term
- Ex.
- Ex. 4
- Ex. y
- Ex.

- Binomial – 2 terms
- Ex.
- Ex.

- Trinomial – 3 terms
- Ex.
- Ex.

- Coefficient – A number that is multiplied by a variable.
- Ex. : the 24 is the coefficient
- Ex. -5x: the -5 is the coefficient

- Variables with Negative Exponents Become Fractions with Variables in the Denominator:
- These are NOT monomials, and not part of polynomials
- Ex. is NOT a monomial

- Square Roots of Variables Become Variables to Fractional Exponents:
- These are NOT monomials, and not part of polynomials
- Ex. is NOT a monomial

- Polynomials in Standard Form are Written in Descending Order
- Ex.

- Constants – A number times x to the zero power
- Ex:
- Ex:
- Use whatever variable is being used in the rest of the polynomial

**Notes**

- Polynomial Conditions – ALL of the conditions below MUST be met for the algebraic expression to be a polynomial:
- Not all algebraic expressions are polynomials.
- An expression is a polynomial if and only if it meets all of the following conditions:
- There are no fractions that contain variables in the denominator.
- There are only whole number exponents (0, 1, 2, …) — no fractional or negative exponents.
- There are no radical expressions that contain variables.
- All coefficients are real, though they may be fractions, radicals, or irrational numbers.

- Closed Systems and Polynomials
- Closed System – Adding, subtracting, or multiplying polynomials will result in another polynomial.
- Not a Closed System – Dividing polynomials may or may not result in a polynomial.
- If the quotient has a variable in the denominator of a fraction, it is NOT a Closed System.

- Adding Polynomials
- Combine like terms
- Put the terms in descending order (from greatest to smallest exponent values)

- Subtracting Polynomials
- Distribute the -1 before you combine like terms.

- Multiplying Binomials Using FOIL

- FOIL: Special Cases

- Multiplying Polynomials
- Distribute the smaller polynomial over the larger polynomial.
- Example:

- Multiplying Polynomials Vertically

- Multiplying Trinomials

- Dividing Polynomials
- It is possible to have a remainder when dividing polynomials.
- The degree of the remainder will always be less than the degree of the divisor.