5.1 – Polynomial Basics

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. $2^5=2 \cdot 2\cdot 2\cdot 2\cdot 2$
• 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. $\frac{2}{3}y^3-y^2+15$ is a polynomial.
• Ex. $3x^2+\frac{5}{x}-2$ is a NOT a polynomial because there is a variable (x) in the denominator of the fraction.

Review

• Nomial – Term
• Terms are separated by plus and minus signs.
• Monomial – 1 term
• Ex. $8x^2$
• Ex. 4
• Ex. y
• Ex. $p^4$
• Binomial – 2 terms
• Ex. $8x^2+1$
• Ex. $x-1$
• Trinomial – 3 terms
• Ex. $8x^2+3x-1$
• Ex. $4a^4-10x+16$

• Coefficient – A number that is multiplied by a variable.
• Ex. $24x^8$: the 24 is the coefficient
• Ex. -5x: the -5 is the coefficient

• Variables with Negative Exponents Become Fractions with Variables in the Denominator: $n^{-a}=\frac{1}{n^a}$
• These are NOT monomials, and not part of polynomials
• Ex. $3x^{-4}$ is NOT a monomial

• Square Roots of Variables Become Variables to Fractional Exponents: $\sqrt{x}=x^{\frac{1}{2}}$
• These are NOT monomials, and not part of polynomials
• Ex. $2 \sqrt{x}$ is NOT a monomial

• Polynomials in Standard Form are Written in Descending Order
• Ex. $5x^5+13x^4-x^3-2x^2+6x-10$

• Constants – A number times x to the zero power
• Ex: $4=4x^0$
• Ex: $-15=-15a^0$
• 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.

• 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.