Warm Up Simplify the following x + 2x x + 2 – 3x Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 1

Section 3: Perform arithmetic operations on polynomials

Essential Question: How can you perform operations on polynomials? Are polynomials closed under addition, multiplication, division, and subtraction? How can polynomials be used to express realistic situations?

Perform arithmetic operations on polynomials  MCC9-12.A.APR.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. (Focus on polynomial expressions that simplify to forms that are linear or quadratic in a positive integer power of x.)

1. Degree (of a polynomial): the highest degree of the polynomial : is the highest degree of its terms when the polynomial is expressed in its canonical form 2. Expression: A mathematical phrase involving at least one variable and sometimes numbers and operation symbols. 3. Monomial expression: An Algebraic expression with one term. 4. Binomial expression : An algebraic expression with two unlike terms. 5. Trinomial expression: An algebraic expression with three unlike terms.

6. Polynomial expression : An expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents. 7. Rational expression: A quotient of two polynomials with a non ‐ zero denominator. 8. Polynomial function A polynomial function is defined as a function, f(x)=, where the coefficients are real numbers. 9. Standard Form of a Polynomial: To express a polynomial by putting the terms in descending exponent order. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9

POLYNOMIAL DEGREE TERMS

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 Polynomials can not have fractions in their exponents, or negative exponents !!!!!!!

POLYNOMIALS MONOMIALS (1 TERM) BINOMIALS (2 TERMS) TRINOMIALS (3 TERMS) When you have just a number it is called a constant Anything larger than 3 terms is only called a polynomial

Polynomials 13 A polynomial of two terms is a binomial. A polynomial of three terms is a trinomial. 7xy 2 + 2y 8x xy + 2y 2 The constant term is 15. The degree is 3. The leading coefficient is 6. The leading coefficient of a polynomial is the coefficient of the variable with the largest exponent. 6x 3 – 2x 2 + 8x + 15 The constant term is the term without a variable.

Ex. 1: 5x 2 + 2x – 4 Ex. 2: 3a 3 + 2a Ex. 3: 5mn 2 Ex. 4: 3x 2 Ex. 5: 4x 2 – 7x Ex. 6: -9x 2 + 2x – 5 Ex. 7: 5ab 2 Ex. 8: -9a 2 bc 3 – 2ab 4 TRINOMIAL BINOMIAL MONOMIAL BINOMIAL TRINOMIAL MONOMIAL BINOMIAL

Arranging Polynomials 15 linearf (x) = mx + bone f (x) = ax 2 + bx + c, a  0quadratictwo cubicthreef (x) = ax 3 + bx 2 + cx + d, a  0 Degree Function Equation Common polynomial functions are named according to their degree. The degree of a polynomial is the greatest of the degrees of any of its terms. The degree of a term is the sum of the exponents of the variables. Examples: 3y 2 + 5x x 5 y + 3x 3 + 2y 2 degree 2 degree 6 Quartic f (x) = ax 4 + bx 3 + cx 2 + dx + e, a  0 four Quintic five Anything higher than five you say the number

NAMING BY THE DEGREE The __________ of a polynomial is the exponent of the term with the greatest exponent(s). DEGREE Find the degree of each polynomial below. Ex. 1: 5x + 9x 2 Degree: Ex. 2: 3x 3 + 5x – x 2 Degree: Ex. 3: -4x + 7 Degree: Ex. 4: -x 4 + 2x 2 + 5x 3 – x Degree: BINOMIAL TRINOMIAL BINOMAL POLYNOMIAL

The degree of a monomial is the sum of the exponents. Ex. 5: 5xy + 9x 2 y 3 Degree: Ex. 6: 3x 3 y 5 + 5xy – x 2 y Degree: Ex. 7: -4xy + 7y 3 Degree: Ex. 8: -x 4 y + 2x 2 y 5 Degree: 5 ADD 2 & 3 ADD 3 & ADD 2 & 5 7 BINOMIAL TRINOMIAL BINOMIAL

Writing Polynomials in standard form Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 18 1) Write the term with the highest exponent first 2) Write the terms with lower exponents in descending order 3) Remember that a variable with no exponent has an understood exponent of 1 4) A constant term (a number with no variable) always goes last. Here's an example: 1) 6y 3 +4y 5 -2y 2 -6y+8y 4 +7

Add Polynomials 19 To add polynomials, combine like terms. Examples: 1. Add (5x 3 + 6x 2 + 3) + (3x 3 – 12x 2 – 10). Use a horizontal format. (5x 3 + 6x 2 + 3) + (3x 3 – 12x 2 – 10) = (5x 3 + 3x 3 ) + (6x 2 – 12x 2 ) + (3 – 10) Rearrange and group like terms. = 8x 3 – 6x 2 – 7 Combine like terms.

Additive Inverse 20 To subtract two polynomials distribute the negative to the polynomial You are subtracting and then add the polynomials together Example: (4x 2 – 5xy + 2y 2 ) – (– x 2 + 2xy – y 2 ). (4x 2 – 5xy + 2y 2 ) – (– x 2 + 2xy – y 2 ) = (4x 2 – 5xy + 2y 2 ) + (x 2 – 2xy + y 2 ) = (4x 2 + x 2 ) + (– 5xy – 2xy) + (2y 2 + y 2 ) = 5x 2 – 7xy + 3y 2 Rewrite the subtraction as the addition of the additive inverse. Rearrange and group like terms. Combine like terms.

Functional Notation 21 Let P(x) = 2x 2 – 3x + 1 and R(x) = – x 3 + x + 5. Examples: 1. Find P(x) + R(x). P(x) + R(x) = (2x 2 – 3x + 1) + (– x 3 + x + 5) = – x 3 + 2x 2 + (– 3x + x) + (1 + 5) = – x 3 + 2x 2 – 2x If D(x) = P(x) – R(x), find D(– 2). P(x) – R(x) = (2x 2 – 3x + 1) – (– x 3 + x + 5) = (2x 2 – 3x + 1) + ( x 3 – x – 5) = x 3 + 2x 2 – 4x – 4 D(– 2) = (– 2) 3 + 2(– 2) 2 – 4(– 2) – 4 = 4

Multiplication of Polynomials 22 To multiply a polynomial by a monomial, use the distributive property and the rule for multiplying exponential expressions. Examples: 1. Multiply: 2x(3x 2 + 2x – 1). = 6x 3 + 4x 2 – 2x 2. Multiply: – 3x 2 y(5x 2 – 2xy + 7y 2 ). = – 3x 2 y(5x 2 ) – 3x 2 y(– 2xy) – 3x 2 y(7y 2 ) = – 15x 4 y + 6x 3 y 2 – 21x 2 y 3 = 2x(3x 2 ) + 2x(2x) + 2x(–1)

Multiplication using the Distributive Property 23 To multiply two polynomials, apply the distributive property. Example: Multiply: (x – 1)(2x 2 + 7x + 3). = (x – 1)(2x 2 ) + (x – 1)(7x) + (x – 1)(3) = 2x 3 – 2x 2 + 7x 2 – 7x + 3x – 3 = 2x 3 + 5x 2 – 4x – 3

FOIL 24 To multiply two binomials use a method called FOIL, which is based on the distributive property. The letters of FOIL stand for First, Outer, Inner, and Last. 1. Multiply the first terms. 3. Multiply the inner terms. 4. Multiply the last terms. 5. Add the products. 2. Multiply the outer terms. 6. Combine like terms.

Example: Multiply 25 Examples: 1. Multiply: (2x + 1)(7x – 5). = 2x(7x) + 2x(–5) + (1)(7x) + (1)(– 5) = 14x 2 – 10x + 7x – 5 2. Multiply: (5x – 3y)(7x + 6y). = 35x xy – 21yx – 18y 2 = 14x 2 – 3x – 5 = 35x 2 + 9xy – 18y 2 = 5x(7x) + 5x(6y) + (– 3y)(7x) + (– 3y)(6y) First Outer Inner Last First Outer Inner Last

Special Products 26 Examples: 1. (3x + 2)(3x – 2) (a + b)(a – b) = a 2 – b 2 = (3x) 2 – (2) 2 = 9x 2 – 4 2. (x + 1)(x – 1) = (x) 2 – (1) 2 = x 2 – 1 The multiply the sum and difference of two terms, use this pattern: = a 2 – ab + ab – b 2 square of the first term square of the second term

Square of a Binomial 27 Examples: 1. Multiply: (2x – 2) 2. (a + b) 2 = (a + b)(a + b) = a 2 + 2ab + b 2 = (2x) 2 + 2(2x)(– 2) + (– 2) 2 = 4x 2 – 8x Multiply: (x + 3y) 2. = (x) 2 + 2(x)(3y) + (3y) 2 = x 2 + 6xy + 9y 2 = a 2 + ab + ab + b 2 To square a binomial, use this pattern: square of the first term twice the product of the two termssquare of the last term

Example: Word Problem 28 Example: The length of a rectangle is (x + 5) ft. The width is (x – 6) ft. Find the area of the rectangle in terms of the variable x. A = L · W = Area x – 6 x + 5 L = (x + 5) ft W = (x – 6) ft A = (x + 5)(x – 6 ) = x 2 – 6x + 5x – 30 = x 2 – x – 30 The area is (x 2 – x – 30) ft 2.

Notice Polynomials are closed under addition, subtraction, multiplication Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 29

EOCT Review Question A train travels at a rate of (4x + 5) miles per hour. How many miles can it travel at that rate in (x – 1) hours? a.) 3x – 4 miles b.) 5x – 4 miles c.) 4x 2 + x – 5 miles d.) 4x 2 – 9x – 5 miles Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 30

Homework Performing arithmetic operations on polynomials worksheet Naming Polynomials Coach book Pages 180 – 181 (all) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 31