Mathematics for Engineering Science and Technology PDF

Mathematics for Engineering Science and Technology COURSE CODE: MATH 0 PROGRAM NAME: BACHELOR OF SCIENCE IN MECHANICAL ENGINEERING (BSME) CREDIT UNITS : 1 UNIT (LAB) Concept of a Polynomial Concept of a Polynomial Polynomial is made up of two terms, namely Poly (meaning “many”) and Nom...

Mathematics for Engineering Science and Technology COURSE CODE: MATH 0 PROGRAM NAME: BACHELOR OF SCIENCE IN MECHANICAL ENGINEERING (BSME) CREDIT UNITS : 1 UNIT (LAB) Concept of a Polynomial Concept of a Polynomial Polynomial is made up of two terms, namely Poly (meaning “many”) and Nominal (meaning “terms.”). A polynomial is defined as an expression which is composed of variables, constants and exponents, that are combined using the mathematical operations such as addition, subtraction, multiplication and division (No division operation by a variable). Based on the numbers of terms present in the expression, it is classified as monomial, binomial, and trinomial. Examples of constants, variables and exponents are as follows: Constants. Example: 1, 2, 3, etc. Variables. Example: g, h, x, y, etc. Exponents: Example: 5 in x5 etc. Definition of terms  Variable – is an object or symbol that changes its value in a particular problem or discussion.  Constant – is an object or symbol that does not change its value in a particular problem or discussion.  Algebraic expression – is a constant, a variable or a combination of constants and variables together with the operations of addition, subtraction, multiplication, division, raising to powers or extraction of roots.  Terms – consist of a variable and constant separated by + or – sign.  Polynomial – is an expression which is composed of terms separated by the fundamental operations in Algebra.  Monomial – is an algebraic expression consisting of only one term.  Binomial – is an algebraic expression which consists of two unlike terms.  Trinomial – is an algebraic expression which consists of three unlike terms.  Multinomial – is an algebraic expression which consists of four or more unlike terms.  Literal coefficient – is the symbol or letter in a particular expression.  Numerical coefficient – is a numeral/number in a particular expression.  Degree of a term – is the sum of the exponents of its variables. Degree of a Polynomial – is the highest exponent found in any non–zero term of the polynomial. Kinds of Polynomials Degree of Polynomials Operations on Polynomials Addition A. Addition of Monomials Procedure 1. To add similar or like terms, find the algebraic sum of the numerical coefficients and prefix it to their common literal factor. Note: The Addition Laws of Signed Numbers a. In adding numbers having the same signs, find their sum and bring down their common sign. b. In adding numbers having different signs, find their difference and bring down the sign of the number with the greater numerical value. 2. To add dissimilar or unlike terms, their sum cannot be written as a single term, instead they can only be indicated. Operations on Polynomials Addition B. Addition of Polynomials There are two ways of adding polynomials: 1. horizontal form 2. vertical form In adding polynomials in horizontal form, simply combine similar terms Operations on Polynomials Addition B. Addition of Polynomials There are two ways of adding polynomials: 1. horizontal form 2. vertical form In adding polynomials in horizontal form, simply combine similar terms Operations on Polynomials Operations on Polynomials  Subtraction  A. Subtraction of Monomials  Procedure  1. To subtract similar or like terms, change the sign of the subtrahend and proceed as in algebraic addition.  2. The subtraction of dissimilar or unlike terms can only be indicated. However, change the sign of the subtrahend. Operations on Polynomials Subtraction B. Subtraction of Polynomials Procedure To subtract polynomials in horizontal form, change the sign of each term of the second polynomial and proceed to addition Operations on Polynomials Operations on Polynomials Operations on Polynomials Operations on Polynomials Operations on Polynomials A. Multiplication of a Monomial by Another Monomial To multiply a monomial by another monomial, find the product of their numerical coefficients and add the exponents of the same variable. Index laws are to be applied. Operations on Polynomials B. Multiplication of a Polynomial by a Monomial Procedure 1. To multiply a polynomial by a monomial, multiply each term of the polynomial by the monomial using the distributive law of multiplication. 2. If possible, arrange the answer in ascending or descending powers of one of the letters. Example Operations on Polynomials C. Multiplication of a Polynomial by Another Polynomial Two methods may be used in this process: Distributive law of multiplication and the long method. In multiplying polynomials, always arrange the given expressions in either descending or ascending order. Using the distributive law of multiplication Distribute or multiply each term of the multiplier by the given multiplicand Example Operations on Polynomials Operations on Polynomials Using the Long Method The process of multiplying polynomials using the long method, is similar to the process used in arithmetic; the only difference is, we start at the left side of the expression, going to the right side. Operations on Polynomials Division ▪ Division of a Polynomial by a Monomial In dividing a polynomial by a monomial, find the quotient of their numerical coefficients and apply the exponential law of division. Operations on Polynomials Division of a Polynomial by Another Polynomial In dividing a polynomial by another, follow the following procedures: 1. To divide polynomials, arrange the terms of both the dividend and divisor in either ascending or descending powers of some common letter. 2. Divide the first term of the dividend by the first term of the divisor and write the result as the first term of the quotient. 3. Multiply the first term of the quotient by the divisor and write this product under the dividend, keeping similar terms under each other. Then subtract the product from the dividend. The difference will be the new dividend. 4. Repeat steps 2 and 3 until you obtain the final answer or the quotient. (Note: The answer is final when the exponent of the dividend is less than the exponent of the divisor). 5. Check your answer by multiplying the quotient by the divisor plus the remainder. Operations on Polynomials Operations on Polynomials To divide a polynomial by a binomial of the type 𝑥 – 𝑟, write the numerical coefficients Synthetic Division: of the terms of the given polynomial arranged according to descending powers of 𝑥, be In the first line, write the numerical coefficients of the terms thus 𝑎0, 𝑎1, 𝑎2, ⋯ 𝑎𝑛. sure to supply each missing power with zero as its coefficient. Write 𝑎0 in the first place in the third line. Multiply this sum by 𝑟, replace the product in the second line under 𝑎2 and add. Continue until the last sum in the third line is obtained. This powers of 𝑥 in the quotient in descending order. is the remainder and the preceding numbers from left to right are the coefficients of the Operations on Polynomials Symbols of Grouping (Signs of Aggregations) Grouping symbols are often necessary to express clearly the meaning of certain expressions and also to indicate the order in which the operations are to be performed. Operations on Polynomials End of Topic 01