Transcript PowerPoint Presentation 12: Algebra

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:
•
2
2
–14a b
+
2
2
(–6a b )
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
4x  6x  15xy
2
2
  9x  x
 2y  5 y
2
2
5x  7x  15xy  2y  5y Ans
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 +
2
2a )
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 (-20a x y ) ÷ (-2ax )
3 5 2
•
•
•
Divide the numerical coefficients
-20 / -2 = 10
Subtract the exponents
a3 – 1= a2
x5 – 2 = x3
y2 = y2
Combine numerical and literal factors
(-20a3x5y2) ÷ (-2ax2) = 10a2x3y2
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 y z)
2 4
•
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:
•
•
27ab c
6 2
Determine the root of the numerical coefficient
3
•
3
27  3
Divide the exponent of the literal factors by the index
3
a  3a
3
b 6  b 6 3  b 2
3
c  c
3
2
Combine
3
2
27ab 6c 2  3b 2 3 ac 2 Ans
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