4.1 Introduction to Polynomials Objectives: • Classify polynomials by the number of terms. • Identify the Leading coefficient of a polynomial. • Identify the degree of a polynomial. • Write polynomials in Standard Form. • Add and Subtract polynomials.

Concept: Classifying Polynomials a number x, y Term: An expression that is a constant, a variable, or a product of a constant and variable(s) that are raised to whole number powers. Multiply a number and a variable The exponent can’t be negative! The exponent can’t have fractions!

Concept: Classifying Polynomials cont… Polynomials are classified by the number of their terms: Monomial: 1 term 2 terms Binomial: 3 terms Trinomial: Polynomial: 4 or more terms

Concept: Coefficient of a term The coefficient of a term is the numerical factor of the term. Ex: Given The coefficient is 4 Copyright © 2011 Pearson Education, Inc.

Concept: Degree of a term The degree of a term is the sum of the exponents of all variables in a term. Ex: Given The exponents of this term are 2 and 3 so the degree of this term is 5 2 + 3 = 5 Copyright © 2011 Pearson Education, Inc.

Got it? Let’s see! Identify the coefficient and degree of term: C: 8

Concept: Degree of a polynomial The degree of a polynomial is the same as the greatest degree of any of the terms in the polynomial. Copyright © 2011 Pearson Education, Inc.

Got it? Let’s see! Identify the type of polynomial and the degree: Monomial D: 3 Binomial D: 2 When an equation in one variable is solved the answer is a point on a line. Trinomial D: 3 Polynomial D: 3 Polynomial D: 4 Not a polynomial

Concept: Evaluate Polynomials To evaluate a polynomial for given values of the variables, you will substitute the values into the polynomial and then simplify. Ex: Evaluate 2x2 + 3xy3 for x = 2 and y = 3 2(2)2 + 3(2)(3)3 2(4) + 3(2)(9) 8 + 54 = 62 Copyright © 2011 Pearson Education, Inc.

Got it? Let’s see! Evaluate each of the following: -2(-1)2(4) = -8 (-4)2 – (-4) – 3 = 16 + 4 – 3 = 17 - (-1)2 (2) = - (1)(2) = -2

Concept: Polynomials in Standard Form To write polynomials in Standard Form you just arrange the terms from largest to smallest degree. Ex: 22m3 + 5m2 – 16m4 + 21 – 14m –16m4 + 22m3 + 5m2 – 14m + 21

Got it? Let’s see! Write the polynomial in descending order.

Concept: Leading Coefficient The leading coefficient of the polynomial is the coefficient of the highest degree term. It is called the leading coefficient because when written in Standard Form, the highest degree term will LEAD the polynomial Ex: 22m3– 14m – 16m4 + 21 + 5m2 –16m4 + 22m3 + 5m2 – 14m + 21 Leading Coefficient = –16

Concept: Adding Polynomials Addition is just combining like terms: (4x2 - 2xy + 3y2) + (-3x2 - xy + 2y2) Line up your like terms. 4x2 - 2xy + 3y2 + -3x2 - xy + 2y2 _________________________ x2 - 3xy + 5y2

Rewrite subtraction as adding the opposite. Concept: Subtracting Polynomials Subtraction just adds one extra step: Subtract: (9y –7x + 15a) – (-3y + 8x – 8a) Rewrite subtraction as adding the opposite. (9y – 7x + 15a) + (+ 3y – 8x + 8a) Group the like terms. 9y + 3y - 7x - 8x + 15a + 8a 12y - 15x + 23a

Concept: Evaluating Made Easy Step #1a: Write the terms of the polygon in Standard Form. Ex: Find f(3): Step #1b: Write the terms of the polygon in Standard Form. Ex: Find f(3):

Concept: Setting It Up cont. . . Step #2: Write the value you are using in the box at the left and write down all of the coefficients. Ex: find 5x4 + 0x3 –4x2 +x +6

Step #3: Bring down the first coefficient, (5). Concept: Doing it Step #3: Bring down the first coefficient, (5). Ex:

Step #4: Multiply the first coefficient by r (3*5). Concept: Doing it cont… Step #4: Multiply the first coefficient by r (3*5). Ex:

Concept: Doing it cont… Step #5: Add the column straight down Add the column Ex:

Concept: Doing it cont… Ste p #6: Multiply the sum (15) by r. (15 • 3 = 45) and place this number under the next coefficient. Then add the column again. Ex:

Concept: Doing it cont… Step #7: Repeat step #6 until you run out of …........... coefficients..

The bottom row of numbers in the division problem was: 5 15 41 124 378 Concept: Writing the answer cont… The bottom row of numbers in the division problem was: 5 15 41 124 378 The value is the last number in the bottom row. f (3) = 378

Find f(-3): f (x)= 2x 3 + x 2 – 8x + 5 Concept: Got it? Let’s see! Find f(-3): f (x)= 2x 3 + x 2 – 8x + 5 SOLUTION: –3 2 1 –8 5 –6 15 –21 2 –5 7 –16 f (–3) = –16 ANSWER:

Homework: PM 4.1

Classify the expression a) Monomial b) Binomial c) Trinomial d) None of these Copyright © 2011 Pearson Education, Inc. 5.2

Classify the expression a) Monomial b) Binomial c) Trinomial d) None of these Copyright © 2011 Pearson Education, Inc. 5.2

Evaluate when x = –3. a) –118 b) –10 c) 10 d) 134 5.2 Copyright © 2011 Pearson Education, Inc. 5.2

Evaluate when x = –3. a) –118 b) –10 c) 10 d) 134 5.2 Copyright © 2011 Pearson Education, Inc. 5.2

Identify the degree of the polynomial. b) 5 c) 6 d) 7 Copyright © 2011 Pearson Education, Inc. 5.2

Identify the degree of the polynomial. b) 5 c) 6 d) 7 Copyright © 2011 Pearson Education, Inc. 5.2

Identify the leading coefficient. b) 5 c) 6 d) 7 Copyright © 2011 Pearson Education, Inc. 5.2

Identify the degree of the polynomial. b) 5 c) 6 d) 7 Copyright © 2011 Pearson Education, Inc. 5.2