Unit #4 Polynomials

The degree of a monomial is the sum of the exponents of the variables. A monomial is the product of a constant times a variable raised to a nonnegative integer power. The degree of a monomial is the sum of the exponents of the variables. The coefficient is the real number of a term.

A polynomial is a monomial or the sum of monomials. The degree of a polynomial is the monomial with the highest degree.

Variables with negative exponents Polynomials cannot have the following: Variables with negative exponents Variable exponents Variables under the radical sign

State the degree of the polynomial

Naming polynomials by the number of terms # Name Example 1 Monomial 4x 2 Binomial 2x + 3 3 Trinomial 4a² – 7a + 2 4 4th Term 4x – 2y + 9z – 6 5 5th Term x³ – 4x²y + 7xy² + y³

Naming polynomials by the degree Names may only be used when the polynomial contains one variable Example 1 Linear 4x 2 Quadratic 2a² + 9 3 Cubic b³ – 4b + 6 4 Quartic 5 Quintic

Simplify the following expressions

Simplify the following expressions

Simplify the following expressions Homework Pages 33 – 34 #30 – 60 Even Page 62 #61 – 68 All

Double Header in Baseball Game Probability Winning Polynomial Form 1 1/3 2 5/8 What is the probability of 1 win and 1 loss ? What is the probability of 2 wins ? What is the probability of 2 losses ?

Three Game Series in Baseball Probability Winning Polynomial Form 1 2/3 2 1/5 3 2/7 What is the probability of 1 win and 2 losses ? What is the probability of 2 wins and 1 loss ? What is the probability of 3 wins ?

2w + l, 1w + 4l, 2w + 5l

Two Game Series in Hockey Prob Win Prob Loss Prob Tie Poly Form 1 3/10 5/10 2/10 2 2/5 1/5 What is the probability of 1 win and 1 loss ? What is the probability of 1 win and 1 tie ?

Binomial Theorem

Pascal’s Triangle 1 2 3 4 6 5 10 15 20 7 21 35

Binomial Theorem Notice each expression has n + 1 terms The degree of each term is equal to n The exponent of each a decreases by 1 and the exponent of each b increases by 1 for each succeeding term in the series The coefficients come from Pascal’s Triangle In subtraction alternate signs starting with positive then negative

Binomial Theorem Write the general rule for the binomial using Pascal’s Triangle Substitute into the general rule Simplify your expression

Expand each of the following using the Binomial Theorem and Pascal’s Triangle

Factorial If n > 0 is an integer, the factorial symbol n! is defined as follows: n! = n(n – 1) •… • 3 • 2 • 1 if n > 2 0! = 1 and 1! = 1 4! = 4 • 3 • 2 • 1 = 24 6! = 6 • 5 • 4 • 3 • 2 • 1 = 720

Binomial Coefficient

Evaluate each of the following expressions

Expand each of the following using the Binomial Theorem and Factorials

We can use the Binomial Theorem to find a particular term in an expression without writing the entire expansion.

Monomial Division

Polynomial Division List all of the terms with their powers in descending order Replace any missing terms with a zero Divide the polynomial until the degree of the divisor is greater than the degree of the remainder Write the remainder over the divisor

Perform the indicated operation below