Transcript Document

Regents Review #1
Expressions
Simplifying Expressions
What does it mean to simplify an expression?
CARRY OUT ALL OPERATIONS!
PEMDAS is always in effect!
Simplifying Exponential Expressions
1) xy0
x(1)
x
any nonzero number
raised to the zero power
equals 1
2) (2x2y)(4xy3)
8x3y4
multiply coefficients and
add exponents
3) (2x3y5)4
24(x3)4(y5)4
16x12y20
raise each factor to the
power
Simplifying Exponential Expressions
2
4) 4 x y
2
xy
4 xy
5)  x 
 y3 


1
4x
y
divide coefficients
subtract exponents
move negative exponents and
rewrite as positive
2
3
6)
x 6
9
y
9
y
x6
raise numerator and
denominator to the power
of the fraction
4 x3 y
24xy3
1x 2 y 2
6
x2
2
6y
simplify numerator and
denominator coefficients by
dividing by a common factor
Simplifying Exponential Expressions
When simplifying exponential expressions, remember…
1) Use exponent rules to simplify
2) When dividing, all results appear in the
numerator. Change negative exponents to
positive by moving them to the other part of the
fraction
3) No decimals or fractions are allowed in any part of
the fraction
Scientific Notation
Writing numbers in scientific notation
1) 345,000,000 = 3.45  108
2) 0.0000109 = 1.09  10-5
Scientific Notation
Multiplying and Dividing Numbers in Scientific
Notation
3)
5
9
4) 5.6  1012
4 10 3.2 10 
4  3.210 10 
9
5
4
12.8 10
4
1.2810 10
1
3
1.2810
4  10
12
5.6 10
 6
4 10
18
1.4  10
6
Polynomials
When adding polynomials, combine like terms!
1) (3x – 2) + (5x – y) + (2x – 4)
3x + 5x + 2x – 2 – 4 – y
10x – 6 – y
Polynomials
When subtracting polynomials, distribute the minus sign before
combining like terms!
2) Subtract 5x2 – 2y from 12x2 – 5y
12x2 – 5y – (5x2 – 2y)
12x2 – 5y – 5x2 + 2y
12x2 – 5x2 – 5y + 2y
7x2 – 3y
Polynomials
When multiplying polynomials, distribute each
term from one set of parentheses to every term in
the other set of parentheses “double distribute”.
3)
3x  2x  4
2
3x  12 x  2 x  8
2
3x  10 x  8
Polynomials
When dividing polynomials, each term in the
numerator is divided by the monomial that
appears in the denominator.
4) 3 x 2 y 4  12x 3 y 2
2
 3x
2
4
3
2
3x y
12x y

2
2
 3x
 3x
4
2
 y  4 xy
Factoring
What does it mean to factor?
Create a “multiplication problem”.
Factoring
There are three ways to factor
1) Factor out the GCF  4x 2  2x  2x(2x 1)
2) AM factoring  x 2  5x  6  ( x  3)(x  2)
3) DOTS  9x2 16y 4  (3x  4 y 2 )(3x  4 y 2 )
Factoring
When factoring completely, factor until you cannot
factor anymore!
1)
2 x 2  10x  12
2)
4 x 2  36 y 2
2( x  5 x  6)
4( x  9 y )
2( x  3)( x  2)
4( x  3 y )( x  3 y )
2
3)
2
2
 x2  x  2
 1( x 2  x  2)  factor out a  1
 1( x  2)(x  1)
Rational Expressions
When simplifying rational expressions (algebraic
fractions), factor and divide out factors that are
common to both the numerator and
denominator.
1)
x2  2x
x( x  2)
x


2
x  3x  2 ( x  2)(x  1) x  1
Rational Expressions
When multiplying, factor and cancel out common factors in the
numerators and denominators of the product.
2)
x 2  x  20
x
( x  5)(x  4)
x
( x  5)




x2  x
x2  2x  8
x( x  1)
( x  4)(x  2) ( x  1)(x  2)
When dividing, multiply by the reciprocal, then factor and divide
out common factors in the numerators and denominators of the
product.
3)
x2  4
x2  6x  8
x2  4
x2 1


 2
2
x 1
x 1
x 1
x  6x  8
( x  2)( x  2)
( x  1)( x  1)
( x  2)( x  1)


( x  1)
( x  2)( x  4)
( x  4)
Rational Expressions
1) When adding and subtracting rational
expressions, find a common denominator
2) Create equivalent fractions using the common
OF ONE
denominator(Multiply by FOOs) FORM
Ex:
2
x
or
2
x
3) Add or subtract numerators and keep the
denominator the same
4) Simplify your final answer if possible
Rational Expressions
4)
Multiply by FOO
4
2

LC D 9 x
9 3x
x 4
2
3 Multiply by FOO
 

x 9 3x
3
4x
6

9x
9x
4x  6
2( 2 x  3)

9 x simplified
9x
Radicals
When simplifying radicals, create a product using
the largest perfect square (4,9,16,25,36,49.64,81,100).
1)
48  16  3  4 3
When multiplying radicals, multiply coefficients
and multiply radicands.
2)
3 2 5 6 
15 12  15  4  3  15  2  3  30 3
Radicals
When dividing radicals, divide coefficients and divide radicands.
3)
6 30
6

2
2 5
30
3
5
6
A fraction is not simplified, if a radical appears in the
denominator!
4)
3
2
3

2
3 2
2
2
 m ultiply by a foo
2
Radicals
When adding or subtracting radicals, simplify all radicals.
If radicals have “like” radicands, then add or subtract
coefficients and keep the radicands the same.
5)
2 32  4 18
2  16  2
24 2


4 3 2
8 2  12 2
4 2
4 9  2
Writing Algebraic Expressions
1) Express the cost of y shirts bought at x dollars each.
xy
2) Express “three times the quantity of 4 less than a
number” as an expression.
3(x – 4)
Evaluating Algebraic Expressions
Evaluate x2 – y when x = -2 and y = -5
x2 – y
(-2)2 – (-5)  always put negative numbersin ( )
4+5
9
Now it’s your turn to review on your own!
Using the information presented today and your review packet,
complete the practice problems in the packet.
Regents Review #2
is
FRIDAY, May 10th
BE THERE!!!!