MATH030 M1 Lesson 2 Operations on Polynomials.pdf

COLLEGE ALGEBRA MATH030 Excellence and Relevance  OPERATIONS ON POLYNOMIAL EXPRESSIONS MODULE 1 LESSON 2 Excellence and Relevance  ORDER OF OPERATIONS RECALL: Correct Order of Oper...

COLLEGE ALGEBRA MATH030 Excellence and Relevance  OPERATIONS ON POLYNOMIAL EXPRESSIONS MODULE 1 LESSON 2 Excellence and Relevance  ORDER OF OPERATIONS RECALL: Correct Order of Operations STEP 1. Start with the innermost PARENTHESES (grouping symbols) and work outward. STEP 2. Perform all indicated EXPONENTS, working from left to right STEP 3. Perform all indicated MULTIPLICATION and DIVISION, working from left to right. STEP 4. Perform all ADDITION and SUBTRACTION, working from left to right. Excellence and Relevance  SAMPLE PROBLEMS 1. Simplify the expression: 2𝑥 + 3 4 − 𝑥 𝑨𝒏𝒔: 𝟏𝟐 − 𝒙 2. Simplify: −{−3 −5 −4𝑥 + 𝑧 − 2 𝑦 + 2𝑧 } 𝑨𝒏𝒔: 𝟔𝟎𝒙 − 𝟔𝒚 − 𝟐𝟕𝒛 3. Simplify: −12 2 + 𝑚 3 − 𝑚 − 𝑚 − 2𝑚 − 6 𝑨𝒏𝒔: 𝟏𝟐𝒎𝟐 − 𝟒𝟖𝒎 + 𝟒𝟖 Excellence and Relevance  TERMINOLOGIES The expressions 3𝑥 2 − 7𝑥 − 1 4𝑦 3 − 𝑦 5𝑧 are all examples of polynomials in one variable. A monomial in one variable, 𝑎𝑥 𝑘 , is the product of a constant and a variable raised to a nonnegative-integer power. The constant 𝒂 is called the coefficient of the monomial, and 𝒌 is called the degree of the monomial. A polynomial is the sum of monomials. The monomials that are part of a polynomial are called terms. ***Like terms – terms that contain the same variables with the same powers. Ex. 3𝑥 and 6𝑥; −5𝑦 2 , 10𝑦 2 and 2𝑦 2 Excellence and Relevance  POLYNOMIALS A polynomial is a mathematical expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, and multiplication. The constants of the polynomials are real numbers, whereas the exponents of the variables are positive integers. Examples: 3𝑥 2 + 2𝑥 + 1 3𝑥 4 − 2𝑥 3 + 5𝑥 2 − 𝑥 + 7 2𝑥 5 + 3𝑥 4 − 2𝑥 2 + 5 3𝑥 3 + 2𝑥 + 5 Excellence and Relevance  NOT POLYNOMIALS These types of expressions do not meet the criteria for polynomials, which require non-negative integer exponents and no division by variables. 1 1. Division by a variable: + 3 𝑥 −2 2. Negative exponents: 𝑥 +4 1 3. Fractional exponents: 𝑥 + 2𝑥 + 3 2 4. Trigonometric functions: sin 𝑥 + 𝑥 5. Exponential functions: 2𝑥 + 𝑥 6. Logarithmic functions: log 𝑥 + 𝑥 Excellence and Relevance  POLYNOMIAL DEGREE The degree of a polynomial is the highest power of the variable in the polynomial. Ex. 4𝑥 3 + 3𝑥 2 + 2𝑥 + 1; the degree is 3 If a polynomial has two or more variables, the degree is the highest sum of the exponents of the variables in any single term. Ex. 2𝑥 3 𝑦 2 + 5𝑥𝑦 4 + 3𝑥 2 + 7; the degree is 5 2𝑥 2 𝑦 3 𝑧 + 4𝑥𝑦 2 𝑧 2 + 3x 3 z + 7; the degree is 6 Excellence and Relevance  ADDITION AND SUBTRACTION OF POLYNOMIALS Addition and subtraction of polynomials are performed by applying the rules of addition and subtraction on like terms. Ex. 5𝑥 2 − 2𝑥 + 3 + (3𝑥 3 − 4𝑥 2 + 7) = 5𝑥 2 − 2𝑥 + 3 + 3𝑥 3 − 4𝑥 2 + 7 = 3𝑥 3 + 𝑥 2 − 2𝑥 + 10 Excellence and Relevance  SAMPLE PROBLEMS Perform the following operations: 1. ADD: 3x 2 + 2xy − y 2, x 2 − 2xy + 6y 2 and 9x 2 + 11xy − 5y 2 𝐀𝐧𝐬: 𝟏𝟑𝐱 𝟐 + 𝟏𝟏𝒙𝒚 2. ADD: 7m2 + 3mn − 5n2 + 3m − 6n + 3 ; 2m2 − mn − m + 5n + 12 ; and 3mn − 5n2 + 4n + 5 𝐀𝐧𝐬: 𝟗𝐦𝟐 + 𝟓𝐦𝐧 − 𝟏𝟎𝐧𝟐 + 𝟐𝐦 + 𝟑𝐧 + 𝟐𝟎 3. Subtract the first polynomial from the second polynomial: 6x + 3xy − 7y and 3x − 2xy + 6y 𝐀𝐧𝐬: −𝟑𝐱 − 𝟓𝐱𝐲 + 𝟏𝟑𝐲 Excellence and Relevance  MULTIPLICATION OF POLYNOMIALS To multiply one polynomial by another, multiply first the polynomial by each term of the second to obtain partial products. Be sure to arrange similar terms in one column. Then add the partial products. This operation can be performed more conveniently if both multiplicand and multiplier are arranged according to the descending (or ascending) powers of some common letters. Ex. −6𝑥 4𝑥 3 − 5𝑥 2 + 𝑥 + 2 = −24𝑥 4 + 30𝑥 3 − 6𝑥 2 − 12𝑥 Excellence and Relevance  MULTIPLICATION OF POLYNOMIALS The method outlines for multiplying polynomials works for all products of polynomials. For the special case when both polynomials are binomials, the FOIL method can also be used. The FOIL method finds the products of the First terms, Outer terms, Inner terms, and Last terms. Excellence and Relevance  SAMPLE PROBLEMS 1. Find the product of 5𝑥 2 + 3𝑥 − 2 and 𝑥 2 − 𝑥 + 1. 𝑨𝒏𝒔: 𝟓𝒙𝟒 − 𝟐𝒙𝟑 + 𝟓𝒙 − 𝟐 2. Find the product of 3𝑥 2 + 2𝑥 − 5 and 𝑥 2 − 4𝑥 + 6. 𝑨𝒏𝒔: 𝟑𝒙𝟒 − 𝟏𝟎𝒙𝟑 + 𝟓𝒙𝟐 + 𝟑𝟐𝒙 − 𝟑𝟎 3. What is the product of 𝑦 10𝑦 3 − 4𝑦 + 6 3𝑦 3 − 4𝑦 − 1 ? 𝑨𝒏𝒔: 𝟑𝟎𝒚𝟕 − 𝟓𝟐𝒚𝟓 + 𝟖𝒚𝟒 + 𝟏𝟔𝒚𝟑 − 𝟐𝟎𝒚𝟐 − 𝟔𝒚 4. What is the product of 12𝑦 3 + 10𝑦 2 − 3𝑦 + 24 −10𝑦 2 − 3𝑦 + 1 ? 𝑨𝒏𝒔: −𝟏𝟐𝟎𝒚𝟓 − 𝟏𝟑𝟔𝒚𝟒 + 𝟏𝟐𝒚𝟑 − 𝟐𝟐𝟏𝒚𝟐 − 𝟕𝟓𝒚 + 𝟐𝟒 5. What is the product of (𝑥 4 − 𝑦 2 )(𝑥 4 + 𝑦 2 )? 𝑨𝒏𝒔: 𝒙𝟖 − 𝒚𝟒 Excellence and Relevance  DIVISION OF POLYNOMIALS Division of polynomials is a mathematical process used to divide one polynomial by another. It is similar in concept to dividing numbers but involves polynomials instead. The goal is to find how many times one polynomial (the divisor) can be multiplied to result in another polynomial (the dividend) and to determine any remainder from this process. The types of polynomial division are long division and synthetic division. Excellence and Relevance  DIVISION OF POLYNOMIALS Steps for long division of polynomials: (2𝑥 3 + 3𝑥 2 − 𝑥 + 5) ÷ (𝑥 − 1) a. Divide the highest degree term: Divide the first term of the dividend 2𝑥 3 by the first term of the divisor 𝑥 , which gives you 2𝑥 2. b. Multiply and subtract: Multiply the entire divisor 𝑥 − 1 by 2𝑥 2 to get 2𝑥 3 − 2𝑥 2. Subtract this from 2𝑥 3 + 3𝑥 2 − 𝑥 + 5 to get 5𝑥 2 − 𝑥 + 5. c. Repeat: Repeat this process using the new dividend 5𝑥 2 − 𝑥 + 5. Divide 5𝑥 2 by 𝑥 to add 5𝑥 to your quotient. Multiply 𝑥 − 1 by 5𝑥 to get 5𝑥 2 − 5𝑥, subtract, and continue. d. Continue until the degree of the remainder is less than the degree of the divisor: The division ends when the remainder has a lower degree than the divisor or when it zeroes out. Excellence and Relevance  SYNTHETIC DIVISION Synthetic division is a shorthand method of dividing polynomials where the divisor is a linear binomial of the form 𝑥 − 𝑐. It simplifies the calculation by using only the coefficients of the polynomials, making the process faster and more efficient than long division. This technique is especially useful for determining if 𝑐 is a root of the polynomial. Excellence and Relevance  SYNTHETIC DIVISION Strategy for Synthetic Division: a. List the coefficients of the polynomial (the dividend). b. Be sure to include zeros for any missing terms in the dividend. c. For dividing by 𝑥 − 𝑐, place 𝑐 to the left. d. Bring the first coefficient down. e. Multiply by 𝑐 and add for each column. f. Read 𝑄 𝑥 and 𝑅 from the bottom row. Excellence and Relevance  SAMPLE PROBLEMS 1. Divide: (2𝑥 4 + 17𝑥 3 + 26𝑥 2 − 21𝑥 + 18) ÷ 𝑥 + 6. 𝑨𝒏𝒔: 𝟐𝒙𝟑 + 𝟓𝒙𝟐 − 𝟒𝒙 + 𝟑 2. Divide (4𝑥 4 − 16𝑥 3 + 𝑥 2 + 24𝑥 + 64) by (2𝑥 − 5). 𝟏𝟏 𝟕𝟑 𝑨𝒏𝒔: 𝟐𝒙𝟑 − 𝟑𝒙𝟐 − 𝟕𝒙 − + 𝟐 𝟐 𝟐𝒙−𝟓 3. Divide (𝑥 3 − 5𝑥 2 + 4𝑥 − 3) by (𝑥 − 2) using synthetic division. 𝟐 𝟕 𝑨𝒏𝒔: 𝒙 − 𝟑𝒙 − 𝟐 − 𝒙−𝟐 4. Is 𝑥 − 1 a factor of 6𝑥 3 − 5𝑥 2 − 4𝑥 + 3? 𝑨𝒏𝒔: Yes! The remainder is zero. Excellence and Relevance  END OF LESSON Excellence and Relevance CREDITS TO: ENGR. FROILAN N. JIMENO II, ECE, ECT