Warm Up Evaluate. 1. – (–2) 4 3. x – 2(3x – 1) 4. 3(y 2 + 6y) –5x + 2 Simplify each expression. – y y

3.1 & 3.2 Identify, evaluate, add, subtract and multiply polynomials. Classify and graph polynomials. Objectives

 Monomial = a number or product of numbers and variables with whole number exponents  Polynomial = a monomial, or a sum or difference of monomials.

The degree is 3 The degree is 2

Identify the degree of each monomial. Example 1: Identifying the Degree of a Monomial A. z 6 B. 5.6 C. 8xy 3 D. a 2 bc 3  The degree of a monomial is the sum of the exponents of the variables.

 Standard Form = the terms of a polynomial are ordered from left to right in descending order, which means from the greatest exponent to the least.  The leading coefficient is the coefficient of the first term. (The term with highest degree.)  Examples: Write each polynomial in standard form.

PolynomialsNot Polynomials |2b 3 – 6b|

 Naming Polynomials by the Number of Terms and by Degree PolynomialDegreeName by Degree # of Terms Name by # of Terms 120Constant1Monomial 8x1Linear1Monomial 2Quadratic2Binomial 3Cubic2Binomial 2Quadratic3Trinomial 4Quartic4Polynomial

Rewrite each polynomial in standard form. Then identify the leading coefficient, degree, and number of terms. Name the polynomial. Example 2: Classifying Polynomials A. 3 – 5x 2 + 4xB. 3x 2 – 4 + 8x 4 –5x 2 + 4x + 3 Write terms in descending order by degree. Leading coefficient: –5 Terms: 3 Name: quadratic trinomial Degree: 2 8x 4 + 3x 2 – 4 Write terms in descending order by degree. Leading coefficient: 8 Terms: 3 Name: quartic trinomial Degree: 4

Check It Out! Example 2 Rewrite each polynomial in standard form. Then identify the leading coefficient, degree, and number of terms. Name the polynomial. a. 4x – 2x 2 + 2b. –18x 2 + x 3 – 5 + 2x –2x 2 + 4x + 2 Leading coefficient: –2 Terms: 3 Name: quadratic trinomial Degree: 2 1x 3 – 18x 2 + 2x – 5 Leading coefficient: 1 Terms: 4 Name: cubic polynomial with 4 terms Degree: 3

To add or subtract polynomials, combine like terms. You can add or subtract horizontally or vertically.

Example 3: Adding and Subtracting Polynomials Add or subtract. Write your answer in standard form. A. (2x – x) + (5x x + x 3 ) (2x – x) + (5x x + x 3 ) Add vertically. Write in standard form. Align like terms. Add. 2x 3 – x + 9 +x 3 + 5x 2 + 7x + 4 3x 3 + 5x 2 + 6x + 13

Example 3: Adding and Subtracting Polynomials Add or subtract. Write your answer in standard form. B. (3 – 2x 2 ) – (x – x) (3 – 2x 2 ) – (x – x) Add the opposite horizontally. Write in standard form. Group like terms. Add. (–2x 2 + 3) + (–x 2 + x – 6) (–2x 2 – x 2 ) + (x) + (3 – 6) –3x 2 + x – 3

Check It Out! Example 3a Add or subtract. Write your answer in standard form. (–36x 2 + 6x – 11) + (6x x 3 – 5) 16x 3 – 30x 2 + 6x – 16 (5x x 2 ) – (15x 2 + 3x – 2) 5x 3 – 9x 2 – 3x + 14

To multiply a polynomial by a monomial, use the Distributive Property and the Properties of Exponents.

Find each product. A. 4y 2 (y 2 + 3) Distribute. B. fg(f 4 + 2f 3 g – 3f 2 g 2 + fg 3 ) 4y 2  y 2 + 4y 2  3 Multiply. 4y y 2 Distribute. Multiply. fg  f 4 + fg  2f 3 g – fg  3f 2 g 2 + fg  fg 3 f 5 g + 2f 4 g 2 – 3f 3 g 3 + f 2 g 4

Find each product. a. 3cd 2 (4c 2 d – 6cd + 14cd 2 ) b. x 2 y(6y 3 + y 2 – 28y + 30) 3cd 2  4c 2 d – 3cd 2  6cd + 3cd 2  14cd 2 12c 3 d 3 – 18c 2 d c 2 d 4 x 2 y  6y 3 + x 2 y  y 2 – x 2 y  28y + x 2 y  30 6x 2 y 4 + x 2 y 3 – 28x 2 y x 2 y

To multiply any two polynomials, use the Box Method to ensure that each term in the second polynomial is multiplied by each term in the first. Keep in mind that if one polynomial has m terms and the other has n terms, then the product has mn terms before it is simplified.

Find the product. (a – 3)(2 – 5a + a 2 ) Step 1: Write polynomials in standard form. Step 2: Create a box with a row/column for each term. Step 3: Multiply Step 4: Combine like terms

(x 2 – 7x + 5)(x 2 – x – 3) Find the product. Step 1: Write polynomials in standard form. Step 2: Create a box with a row/column for each term. Step 3: Multiply Step 4: Combine like terms

Find the product. Step 1: Write polynomials in standard form. Step 2: Create a box with a row/column for each term. Step 3: Multiply Step 4: Combine like terms (x 2 – 4x + 1)(x 2 + 5x – 2)

Find the product. (a + b) 2 (a + b) 3

There are n+1 terms and the powers sum for each term = n.

Expand each expression. A. (k – 5) Identify the coefficients for n = 3, or row 3. [1(k) 3 (–5) 0 ] + [3(k) 2 (–5) 1 ] + [3(k) 1 (–5) 2 ] + [1(k) 0 (–5) 3 ] k 3 – 15k k – 125

Expand each expression. a. (x + 2) Identify the coefficients for n = 3, or row 3. [1(x) 3 (2) 0 ] + [3(x) 2 (2) 1 ] + [3(x) 1 (2) 2 ] + [1(x) 0 (2) 3 ] x 3 + 6x x + 8

b. (x – 4) Identify the coefficients for n = 5, or row 5. [1(x) 5 (–4) 0 ] + [5(x) 4 (–4) 1 ] + [10(x) 3 (–4) 2 ] + [10(x) 2 (–4) 3 ] + [5(x) 1 (–4) 4 ] + [1(x) 0 (–4) 5 ] x 5 – 20x x 3 – 640x x – 1024

Expand each expression. c. (y - 3) 4 d. (4x + 5) 3 y 4 – 12y y 2 – 108y x x x + 125

Throughout this chapter, you will learn skills for analyzing, describing, and graphing higher- degree polynomials. Until then, the graphing calculator will be a useful tool.

Example 5: Graphing Higher-Degree Polynomials on a Calculator Graph each polynomial function on a calculator. Describe the graph and identify the number of real zeros. A. f(x) = 2x 3 – 3x From left to right, the graph increases, then decreases, and increases again. It crosses the x-axis 3 times, so there appear to be 3 real zeros. From left to right, the graph alternately increases and decreases, changing direction 3 times and crossing the x-axis 4 times; 4 real zeros. B.

Check It Out! Example 5 Graph each polynomial function on a calculator. Describe the graph and identify the number of real zeros. a. f(x) = 6x 3 + x 2 – 5x + 1b. f(x) = 3x 2 – 2x + 2 From left to right, the graph increases, decreases slightly, and then increases again. It crosses the x-axis 3 times, so there appear to be 3 real zeros. From right to left, the graph decreases and then increases. It does not cross the x-axis, so there are no real zeros.

Check It Out! Example 5 Graph each polynomial function on a calculator. Describe the graph and identify the number of real zeros. c. g(x) = x 4 – 3d. h(x) = 4x 4 – 16x From left to right, the graph decreases and then increases. It crosses the x-axis twice, so there appear to be 2 real zeros. From left to right, the graph alternately decreases and increases, changing direction 3 times. It crosses the x-axis 4 times, so there appear to be 4 real zeros.

HA2 Homework 3.1, p 154:14, (odd) 36-39, 50, 51 a, c, e p 157 on a separate sheet 3.2, p 162: 19, 23, 26-29, 34, 58-61

Alg 2 Homework 3.1, p 154: (odd) 3.2, p 162:1-25 (every other odd), 31, 33