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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
5 + 10 = 15
• Leave the literal factor unchanged 5x + 10x = 15x
•Example: –14a2b2 + (–6a2b2) • Add the numerical coefficients and leave the literal factor
unchanged–14 + –6 = –20 –14a2b2 + (–6a2b2) = –20a2b2
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) + 5xy2 and (–2x) + 3xy + (–6xy2)
• 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
(4x2 + 6x – 15xy) – (9x2 – x – 2y + 5y2)
• Change the sign of each term in the subtrahend
–9x2 + x + 2y – (5y2)
• Follow the procedure for addition of signed numbers
2
2 24 6 159 2 5x x xyx x y y
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 (2xy2)(-3x2y3)
• Multiply the numerical coefficients following the procedure for multiplication of signed numbers(2)(-3) = -6
• Add the exponents of the same literal factors(x)(x2) = x1+2 = x3 and (y2)(y3) = y2+3 = y5
• Show the product of coefficients and literal factors(2xy2)(-3x2y3) = -6x3y5
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 + 2a2)
• Multiply each term of one expressions by each term of the other expression
= 3a(6) + 3a(2a2)
= 18a + 6a3
• Combine like terms; since 18a and 6a3 are unlike terms, they can not be combined
= 18a + 6a3
MULTIPLICATION•Example: (3c + 5d2)(4d2 – 2c)
• Multiply each term of one expressions by each term of the other expression (FOIL method)
3c (4d2) = 12cd2 (F)irst term
3c(–2c) = –6c2 (O)uter term
5d2(4d2) = 20d4 (I)nner term
5d2(–2c) = –10cd2 (L)ast term
• Combine like terms
(3c + 5d2)(4d2 – 2c) = 2cd2 –6c2 + 20d4
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 (-20a3x5y2) ÷ (-2ax2)
• Divide the numerical coefficients-20 / -2 = 10
• Subtract the exponentsa3 – 1= a2
x5 – 2 = x3
y2 = y2
• Combine numerical and literal factors (-20a3x5y2) ÷ (-2ax2) = 10a2x3y2
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: (–4x2y4z)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
(x2y4z)3 = x2(3) + y4(3) + z1(3) = x6y12z3
• Combine
(–4x2y4z)3 = –64x6y12z3
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 + 5b3)2
• Apply the FOIL method
3a(3a) = 9a2 (F)irst term
3a(5b3) = 15ab3 (O)uter term
5b3(3a) = 15ab3 (I)nner term
5b3(5b3) = 25d6 (L)ast term
• Combine
9a2 + 30ab3 + 25d6
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
6 23 27ab c
3 3
3 6 6 3 2
3 32 2
a a
b b b
c c
3 27 3
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: –(7a2 + b – 3) + 12 – (– b + 5)= – 7a2 – b + 3 + 12 + b – 5 = – 7a2 + 10
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 – 2b2) • Multiply
4b(5 + a – 2b2) = 20b + 4ab – 8b3
• Combine like terms
5b + 20b = 25b
25b + 4ab – 8b3
