Polynomials. 4-4 to 4-8

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1 Polynomials 4-4 to 4-8

2 Learning Objectives 4-4 Polynomials Monomials, binomials, and trinomials Degree of a polynomials Evaluating polynomials functions

3 Polynomials Polynomials are sums of these "variables and exponents" expressions. Each piece of the polynomial, each part that is being added, is called a "term". Polynomial terms have variables which are raised to whole-number exponents (or else the terms are just plain numbers). There are no square roots of variables, no fractional powers, and no variables in the denominator of any fractions.

4 Polynomials A typical polynomial:

5 Polynomials 6x 2 This is NOT a polynomial term......because the variable has a negative exponent. x 2 y x This is NOT a polynomial term... This is NOT a polynomial term......because the variable is in the denominator....because the variable is inside a radical. 4x 2 This IS a polynomial term......because it obeys all the rules.

6 Polynomial Degrees Second-degree polynomial, 4x 2, x 2 9, or ax 2 + bx + c Third-degree polynomial, 6x 3 or x 3 27 Fourth-degree polynomial, x 4 or 2x 4 3x Fifth-degree polynomial, 2x 5 or x 5 4x 3 x + 7

7 Monomial An expression containing only one term is called a monomial. Example: 7, x, 7x, -6x, ab, etc. A monomial is a number, a variable, or the product of a number and one or more variables with whole number exponents. Not a monomial: 8 + x, 2/n, 5 x, a -1, -3x -3, x 2.5

8 Binomial and Trinomial Binomial: An expression containing two terms is called a binomial. Examples: 7x+5, 6y - p Trinomial: An expression containing three terms is called a trinomial. Examples: 2x+3y-4z

9 Monomial, Binomial, and Trinomial Type Definition Example Monomial A polynomial with one term 5x Binomial A polynomial with two terms 5x - 10 Trinomial A polynomial with three terms

10 Evaluating Polynomial Functions Evaluating a polynomial is the same as evaluating anything else: you plug in the given value of x, and figure out what y is supposed to be. Evaluate f(x) = 2x 3 x 2 4x + 2 at f(-3)

11 Examples h( x) x 3 x 2x 3 Find: a) h(0) b) h(-3)

12 Evaluating Polynomial Functions The revenue ($) that a mfg. of desks receives is given by the polynomial function: f(d) = -0.08d d where d is the number of desks. a) Find the total revenue if 625 desks are made. b) Does increasing the number of desks being made to 650 increase the revenue?

13 Section 4.4 Review Polynomials Monomials, binomials, and trinomials Degree of a polynomials Evaluating polynomials functions

14 Section 4.5 Learning Objective Adding and subtracting monomials Adding and subtracting polynomials Adding and subtracting multiples of polynomials An application of adding polynomials

15 Remember Like Terms? 4x and 3 NOT like terms The second term has no variable 4x and 3y NOT like terms The second term now has a variable, but it doesn't match the variable of the first term 4x and 3x 2 NOT like terms The second term now has the same variable, but the degree is different 4x and 3x LIKE TERMS Now the variables match and the degrees match

16 Adding and Subtracting Monomials Step 1: Remove the ( ). Step 2: Combine like terms. Examples: 4ab + (-2ab) = 4ab - (-2ab) = 6x 2 - x 2 =

17 Adding Polynomials Examples: (5p 2 3) + (2p 2 3p 3 ) (4 + 2n 3 ) + (5n 3 + 2)

18 Subtracting Polynomials Examples: (a 3 2a 2 ) - (3a 2 4a 3 ) (4r 3 + 3r 4 ) (r 4 5r 3 )

19 Adding & Subtracting Multiples of Polynomials Example: Add 3(x 2 + 4x) and 2(x 2 4)

20 Application #1 A house is purchased for $105,000 and is expected to appreciate $900 per year, its value y after x years is given by the polynomial function f(x) = 900x + 105,000. a) What is the expected values in 10 years?

21 Application #2 A house second home is purchased for $120,000 and is expected to appreciate $1,000 per year. a) Find a polynomial function that will give the appreciated value y of the house in x years. b) Find the value of this second house after 12 years.

22 Section 4.5 Review Adding and subtracting monomials Adding and subtracting polynomials Adding and subtracting multiples of polynomials An application of adding polynomials

23 Section 4.6 Learning Objectives Multiplying monomials Multiplying a polynomial by a monomial Multiplying a binomial by a binomial The FOIL method Special products Multiplying a polynomial by a binomial Multiplying three polynomials Multiplying binomials to solve equations

24 Multiplying Monomials When multiplying two monomials, multiply the numerical factors and then multiply the variable factors. Example: (5x 2 y 3 )(6x3y 4 ) 30x 3 3y 7

25 Multiplying a Polynomial by a Monomial Use the distributive property to remove parentheses and simplify. Example: 2x 3 (3x 2 5x)

26 Multiplying a Binomial by a Binomial Multiply each term of one binomial by each term of the other binomial and combine like terms. Example: (x + 3)(x + 2) (x + 3y)(2x 5y)

27 The FOIL Method F O I L First terms Outside terms Inside terms Last terms Find the product of (z + 3)(z + 1) One way to keep track of your distributive property is to Use the FOIL method. Note that this method only works on (Binomial)(Binomial).

28 The Vertical Method and Grid Method Multiply: (x + 2)(x + 3) The vertical method: The grid method:

29 Examples Find each product: 1. (3y 2) 2 2. (5t 2u)(2t + 3u)

30 Special Products Square of the sums: (x + y) 2 = X 2 + 2xy +y 2 The square of the differences: (x y ) 2 = X 2 2xy +y 2 Product of the sum and difference of two terms: (a + b)(a b) = a 2 b 2

31 Examples (z + 6) 2 (7x 2) 2 (5m 9n)(5m + 9n)

32 Multiplying a polynomial by a binomial Rule: To multiply one polynomial by another, multiply each term of one polynomial by each term of the other polynomial and combine like terms.

33 Examples (3x + 2)(2x 2 4x + 5) (-2x 2 + 3)(2x 2 4x -1)

34 Multiplying Three Polynomials -2y(y + 3)(3y 2) Solve the equation: (x + 2)(x + 3) = x(x + 7)

35 Dividing by Polynomials Monomials 5p 10 2 pq q b 4

36 Dividing Polynomials by Polynomials Divide x 2 x 2 5x 6 Divide x 1 x 2 9x 10

37 Dividing Polynomials by Polynomials 11x 10x Divide by 5x P339, 22

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