Algebraic Expressions & Polynomials Chapter 5 Sections 5.1-5.3

Copyright © Cengage Learning. All rights reserved. 5.1 Fundamental Operations Copyright © Cengage Learning. All rights reserved.

Fundamental Operations In arithmetic, we perform mathematical operations with specific numbers. In algebra, we perform these same basic mathematical operations with numbers and variables—letters that represent unknown quantities. To begin our study of algebra, some basic mathematical principles that you will apply are listed below. Note that “≠” means “is not equal to.”

Fundamental Operations Basic Mathematical Principles 1. a + b = b + a (Commutative Property for Addition) 2. ab = ba (Commutative Property for Multiplication) 3. (a + b) + c = a + (b + c) (Associative Property for Addition) 4. (ab)c = a(bc) (Associative Property for Multiplication) 5. a(b + c) = ab + ac, or (b + c)a = ba + ca (Distributive Property)

Fundamental Operations 6. a + 0 = a 7. a  0 = 0 8. a + (–a) = 0 (Additive Inverse) 9. a  1 = a 10. a  = 1 (a ≠ 0) (Multiplicative Inverse)

Fundamental Operations In mathematics, letters are often used to represent numbers. Thus, it is necessary to know how to indicate arithmetic operations and carry them out using letters. Addition: x + y means add x and y. Subtraction: x – y means subtract y from x or add the negative of y to x; that is, x + (–y). Multiplication: xy or x  y or (x)(y) or (x)y or x(y) means multiply x by y.

Fundamental Operations Division: x  y or means divide x by y, or find a number z such that zy = x. Exponents: xxxx means use x as a factor 4 times, which is abbreviated by writing x4. In the expression x4, x is called the base, and 4 is called the exponent. For example, 24 means 2  2  2  2 = 16.

Fundamental Operations Order of Operations 1. Perform all operations inside parentheses first. If the problem contains a fraction bar, treat the numerator and the denominator separately. 2. Evaluate all powers, if any. For example, 6  23 = 6  8 = 48. 3. Perform any multiplications or divisions in order, from left to right. 4. Do any additions or subtractions in order, from left to right.

Example 1 Evaluate: 4 – 9(6 + 3)  (–3). = 4 – 9(9)  (–3) = 4 – 81  (–3) = 4 – (–27) = 31 Add within parentheses. Multiply. Divide. Subtract.

Fundamental Operations To evaluate an expression, replace the letters with given numbers; then do the arithmetic using the order of operations. The result is the value of the expression. Evaluate ab/3c + c if a=6 b=10 c= -5 6(10)/3(-5) + (-5)= 60/-15 + (-5)= -4 + (-5)= -9

Copyright © Cengage Learning. All rights reserved. 5.2 Simplifying Algebraic Expressions Copyright © Cengage Learning. All rights reserved.

Simplifying Algebraic Expressions Parentheses are often used to clarify the order of operations when the order of operations is complicated or may be ambiguous. Sometimes it is easier to simplify such an expression by first removing the parentheses—before doing the indicated operations.

Simplifying Algebraic Expressions Two rules for removing parentheses are as follows Removing Parentheses 1. Parentheses preceded by a plus sign may be removed without changing the signs of the terms within. Think of using the Distributive Property, a(b + c) = ab + ac, and multiplying each term inside the parentheses by 1. That is, 3w + (4x + y) = 3w + 4x + y

Simplifying Algebraic Expressions 2. Parentheses preceded by a minus sign may be removed if the signs of all the terms within the parentheses are changed; then the minus sign that preceded the parentheses is dropped. Think of using the Distributive Property, a(b + c) = ab + ac, and multiplying each term inside the parentheses by –1. That is, 3w – (4x – y) = 3w – 4x + y (Notice that the sign of the term 4x inside the parentheses is not written. It is therefore understood to be plus.)

Example 1 Remove the parentheses from the expression 5x – (– 3y + 2z). 5x – (– 3y + 2z) = 5x + 3y – 2z Change the signs of all of the terms within parentheses; then drop the minus sign that precedes the parentheses.

Simplifying Algebraic Expressions A term is a single number or a product of a number and one or more letters raised to powers. The following are examples of terms: 5x, 8x2, – 4y, 15, 3a2b3, t The numerical coefficient is the numerical factor of a term. The numerical factor of the term 16x2 is 16. The numerical coefficient of the term – 6a2b is – 6. The numerical coefficient of y is 1.

Simplifying Algebraic Expressions Terms are parts of an algebraic expression separated by plus and minus signs. For example, 3xy + 2y + 8x2 is an expression consisting of three terms.

Like Terms

Like Terms Terms with the same variables with exactly the same exponents are called like terms. For example, 4x and 11x have the same variables and are like terms. The terms – 5x2y3 and 8x2y3 have the same variables with the same exponents and are like terms. The terms 8m and 5n have different variables, and the terms 7x2 and 4x3 have different exponents, so these are unlike terms.

Example 3 The following table gives examples of like terms and unlike terms.

Like Terms Like terms that occur in a single expression can be combined into one term by combining coefficients (using the Distributive Property). Thus, ba + ca = (b + c)a.

Example 4 Combine the like terms 2x + 3x. 2x + 3x = (2 + 3)x = 5x

Like Terms Some expressions contain parentheses that must be removed before combining like terms. Follow the order of operations. a(b + c) = ab + ac. The Distributive Property is applied to remove parentheses when a number, a letter, or some product precedes the parentheses.

Example 12 Simplify: 3x + 5(x – 3). 3x + 5(x – 3) = 3x + 5x – 15 Apply the Distributive Property by multiplying each term within the parentheses by 5. Combine like terms.

5.3 Addition and Subtraction of Polynomials Copyright © Cengage Learning. All rights reserved.

Addition and Subtraction of Polynomials A monomial, or term, is any algebraic expression that contains only products of numbers and variables, which have nonnegative integer exponents. The following expressions are examples of monomials: 2x, 5, –3b, A polynomial is either a monomial or the sum or difference of unlike monomials. We consider two special types of polynomials.

Addition and Subtraction of Polynomials A binomial is a polynomial that is the sum or difference of two unlike monomials. A trinomial is the sum or difference of three unlike monomials. The following table shows examples of monomials, binomials, and trinomials.

Addition and Subtraction of Polynomials Expressions that contain variables in the denominator are not polynomials. For example, and are not polynomials.

Example 1 Find the degree of each monomial: a. –7m, b. 6x2, c. 5y3, d. 5. a. –7m has degree 1. b. 6x2 has degree 2. c. 5y3 has degree 3. d. 5 has degree 0 The exponent of m is 1. The exponent of x is 2. The exponent of y is 3. 5 may be written as 5x0.

Addition and Subtraction of Polynomials A polynomial is in decreasing order if each term is of some degree less than the preceding term. The following polynomial is written in decreasing order: 4x5 – 3x4 – 4x2 – x + 5 exponents decrease

Addition and Subtraction of Polynomials A polynomial is in increasing order if each term is of some degree larger than the preceding term. The following polynomial is written in increasing order: 5 – x – 4x2 – 3x4 + 4x5 Adding Polynomials To add polynomials, add their like terms. exponents increase

Example 3 Add: (3x + 4) + (5x – 7). (3x + 4) + (5x – 7) = (3x + 5x) + [4 + (–7)] = 8x – 3 Add the like terms.

Addition and Subtraction of Polynomials Subtracting Polynomials To subtract two polynomials, change all the signs of the terms of the second polynomial and then add the two resulting polynomials.

Example 7 Subtract: (5a – 9b) – (2a – 4b). (5a – 9b) – (2a – 4b) = [5a + (–2a)] + [(–9b) + 4b] = 3a – 5b Change all the signs of the terms of the second polynomial and add. Add the like terms.

Group Practice Problems Page 241 3, 4, 5, 7