Section 6.2 and 6.3 Polynomials-- A function with several terms that are added or subtracted

together, such as:• 5x4 + 3x3 – 10x2 + 17x – 9• 10x5 – 9x4 + 13x3 + 23x2 – ½ x + 7

The degree of a polynomial is determined by the term with the largest exponent:

5x4 + 3x3 – 10x2 + 17x – 9is a 4th degree polynomial.

10x5 – 9x4 + 13x3 + 23x2 – ½ x + 7is a 5th degree polynomial.

8x4y2 + 15x3y + 4xy2

this is actually a 6th degree polynomial, because the first term’s exponents combine to 6

More about Polynomials• All exponents must be whole numbers

• f(x) = 5x3 + 2x-1 These are not polynomials• f(x) = 4x2 + 3x

Standard form of a Polynomial – arrange the terms left to right starting with the highest exponent:

7x5 + 10x4 – 12x3 + 17x2 – 9x + 6

10x5 + 7x3 – 2x + 1

More about Polynomials• Polynomials are named by how many terms they have:

• Monomial – one term…..7x• Binomial – two terms…. 3x + 9• Trinomial – three terms… 7x2 + 9x - 2• Polynomial – four or more terms… 4x3 – 8x2 + 6x + 12

Adding and Subtracting Polynomials

• Combine like terms with the same exponents

• Example:(10x5 – 9x4 + 13x3 + 23x2 – 4x + 7) + (5x4 + 3x3 – 10x2 + 17x – 9)

= 10x5 – 4x4 + 16x3 + 13x2 + 13x - 2

Multiplying Polynomials

• Use the distributive property, and remember your properties for exponents from Chapter 6.1.

5x6 (11x3 – 6x2) = 55x9 – 30x8

-7x4y2 (2x8y5 + 9x3y4) = -14x12y7 – 63x7y6

Multiplying Polynomials

• Use FOIL for two Binomials(2x2 + 3) (5x4 + 9x) = 10x6 + 18x3 + 15x4 + 27x

Put in correct order: 10x6 + 15x4 + 18x3 + 27x

Multiplying Polynomials

(2x + 3) (5x + 7) (2x – 4) Use FOIL for the first two:= 10x2 + 29x + 21

Then multiply: (10x2 + 29x + 21) (2x – 4)

= 20x3 – 40x2 + 58x2 – 116x + 42x – 84Simplify: = 20x3 + 18x2 – 74x - 84

Multiplying Larger Polynomials

(5x + 6) (3x2 + 7x + 4) Similar to using FOIL

Multiply 5x by each term in the 2nd polynomial15x3 + 35x2 + 30x

Then multiply 6 by each term in the 2nd polynomial.18x2 + 42x + 24Then combine the like terms and put in order:15x3 + 53x2 + 72x + 42

Multiplying Larger Polynomials

(9x3 + 6x2 + 2) (2x2 -3x + 1)

Multiply each term in the first equation by each term in the second equation

= 18x5 – 27x4 + 9x3 + 12x4 -18x3 + 6x2 +4x2 – 6x + 2Combine like terms: = 18x5 – 15x4 – 9x3 + 10x2 – 6x + 2

Practice

• Page 341, #32,38,44