PRESENTATION 12 Basic Algebra

BASIC ALGEBRA DEFINITIONS A term of an algebraic expression is that part of the expression that is separated from the rest by a plus or minus sign A factor is one of two or more literal and/or numerical values of a term that are multiplied A numerical coefficient is the number factor of a term The letter factors of a term are the literal factors

BASIC ALGEBRA DEFINITIONS Like terms are terms that have identical literal factors Unlike terms are terms that have different literal factors or exponents

ADDITION Only like terms can be added. The addition of unlike terms can only be indicated Procedure for adding like terms: Add the numerical coefficients, applying the procedure for addition of signed numbers Leave the variables unchanged

ADDITION Example: Add 5x and 10x Add the numerical coefficients = 15 Leave the literal factor unchanged 5x + 10x = 15x Example: –14a 2 b 2 + (–6a 2 b 2 ) Add the numerical coefficients and leave the literal factor unchanged –14 + –6 = –20 –14a 2 b 2 + (–6a 2 b 2 ) = –20a 2 b 2

ADDITION Procedure for adding expressions that consist of two or more terms: Group like terms in the same column Add like terms and indicate the addition of the unlike terms

ADDITION Example: Add the two expressions 7x + (–xy) + 5xy 2 and (–2x) + 3xy + (–6xy 2 ) Group like terms in the same column Add the like terms and indicate the addition of the unlike terms

SUBTRACTION Just as in addition, only like terms can be subtracted Each term of the subtrahend is subtracted following the procedure for subtraction of signed numbers

SUBTRACTION Example: Subtract the following expressions (4x 2 + 6x – 15xy) – (9x 2 – x – 2y + 5y 2 ) Change the sign of each term in the subtrahend –9x 2 + x + 2y – (5y 2 ) Follow the procedure for addition of signed numbers

MULTIPLICATION In multiplication, the exponents of the literal factors do not have to be the same to multiply the values Procedure for multiplying two or more terms: Multiply the numerical coefficients, following the procedure for multiplication of signed numbers Add the exponents of the same literal factors Show the product as a combination of all numerical and literal factors

MULTIPLICATION Example: Multiply (2xy 2 )(-3x 2 y 3 ) Multiply the numerical coefficients following the procedure for multiplication of signed numbers (2)(-3) = -6 Add the exponents of the same literal factors (x)(x 2 ) = x 1+2 = x 3 and (y 2 )(y 3 ) = y 2+3 = y 5 Show the product of coefficients and literal factors (2xy 2 )(-3x 2 y 3 ) = -6x 3 y 5

MULTIPLICATION Procedure for multiplying expressions that consist of more than one term within an expression: Multiply each term of one expression by each term of the other expression Combine like terms

MULTIPLICATION Example: 3a(6 + 2a 2 ) Multiply each term of one expressions by each term of the other expression = 3a(6) + 3a(2a 2 ) = 18a + 6a 3 Combine like terms; since 18a and 6a 3 are unlike terms, they can not be combined = 18a + 6a 3

MULTIPLICATION Example: (3c + 5d 2 )(4d 2 – 2c) Multiply each term of one expressions by each term of the other expression (FOIL method) 3c (4d 2 ) = 12cd 2 (F)irst term 3c(–2c) = –6c 2 (O)uter term 5d 2 (4d 2 ) = 20d 4 (I)nner term 5d 2 (–2c) = –10cd 2 (L)ast term Combine like terms (3c + 5d 2 )(4d 2 – 2c) = 2cd 2 –6c d 4

DIVISION Procedure for dividing two terms: Divide the numerical coefficients following the procedure for division of signed numbers Subtract the exponents of the literal factors of the divisor from the exponents of the same letter factors of the dividend Combine numerical and literal factors

DIVISION Example: Divide (-20a 3 x 5 y 2 ) ÷ (-2ax 2 ) Divide the numerical coefficients -20 / -2 = 10 Subtract the exponents a 3 – 1 = a 2 x 5 – 2 = x 3 y 2 = y 2 Combine numerical and literal factors (-20a 3 x 5 y 2 ) ÷ (-2ax 2 ) = 10a 2 x 3 y 2

POWERS Procedure for raising a single term to a power: Raise the numerical coefficients to the indicated power following the procedure for powers of signed numbers Multiply each of the literal factor exponents by the exponent of the power to which it is raised Combine numerical and literal factors

POWERS Example: (–4x 2 y 4 z) 3 Raise the numerical coefficients to the indicated power (–4) 3 = (–4)(–4)(–4) = –64 Multiply the exponents of the literal factors by the indicated powers (x 2 y 4 z) 3 = x 2(3) + y 4(3) + z 1(3) = x 6 y 12 z 3 Combine (–4x 2 y 4 z) 3 = –64x 6 y 12 z 3

POWERS Procedure for raising two or more terms to a power: Apply the procedure for multiplying expressions that consist of more than one term

POWERS Example: (3a + 5b 3 ) 2 Apply the FOIL method 3a(3a) = 9a 2 (F)irst term 3a(5b 3 ) = 15ab 3 (O)uter term 5b 3 (3a) = 15ab 3 (I)nner term 5b 3 (5b 3 ) = 25d 6 (L)ast term Combine 9a ab d 6

ROOTS Procedures for extracting the root of a term: Determine the root of the numerical coefficient following the procedure for roots of signed numbers The roots of the literal factors are determined by dividing the exponent of each literal factor by the index of the root Combine the numerical and literal factors

ROOTS Example: Determine the root of the numerical coefficient Divide the exponent of the literal factors by the index Combine

REMOVAL OF PARENTHESES Procedure for removal of parentheses preceded by a plus sign: Remove the parentheses without changing the signs of any terms within the parentheses Combine like terms Example: – 7x + (–4x + 3y – 2) = –7x – 4x + 3y – 2 = –11x + 3y – 2

REMOVAL OF PARENTHESES Procedure for removal of parentheses preceded by a minus sign: Remove the parentheses while changing the signs of any terms within the parentheses Combine like terms Example: –(7a 2 + b – 3) + 12 – (– b + 5) = – 7a 2 – b b – 5 = – 7a

COMBINED OPERATIONS Expressions that consist of two or more different operations are solved by applying the proper order of operations Example: 5b + 4b(5 + a – 2b 2 ) Multiply 4b(5 + a – 2b 2 ) = 20b + 4ab – 8b 3 Combine like terms 5b + 20b = 25b 25b + 4ab – 8b 3