Download Algebra Standard

ALGEBRA
p. 1
7.2 Combine and simplify algebraic expressions that contain polynomials, rational expressions, radicals, or rational exponents.
7.2 Combine and simplify algebraic expressions that contain polynomials, rational expressions, radicals, positive or negative rational exponents, and logarithmic
expressions.
High School
Community College
EWU
a. Find the sum, difference, or product of two polynomials, then simplify the result.
Core 1
Core 2
SFCC – Beginning & Intermediate Algebra
Perform the operation and simplify:
(a) 12r 4  7r 3  2r 2  5r 4  3r 3  2r 2 1

(b)
(c)
Core 3
hx   6x  3  5x  2
Write x  7x  7 in expanded polynomial form.
Find all possible sums , differences and products of the
four given polynomial functions.
(a) f x   4 x 3  x 2  7 x  17
(b) g x   3 x 2  4 x  7
(c) hx   5x  2
(d)
j x   5 x 4  3 x 2
Integrated 3
Multiply 2 x  3x  5 and add the product to
x 2  3x  5 .
Algebra 2, CPM
Bob, Kris, Janelle, and Pat are in a study team. Bob,
Kris, and Janelle have algebra tiles on their desks. Bob
has two x 2 -tiles, 4 x-tiles, and seven unit tiles. Kris
has one x 2 tile, and five unit tiles; and Janelle has ten
x-tiles, three unit tiles. Pat’s desk is empty. The team
(d)
 
s 1
x  23x  8
2x 2  5q3x  8q

Basic Algebra
Given the polynomial 3x 2  4 x  2 , add it to the
polynomial  x 2  3x  1 , and subtract it from 2x 5 .
3
Which of the following result in a value of 0?
(a)  4x 3  2 x 2 y  1  4x 3  2 x 2 y 1
(b)
SCC – Intermediate Algebra
Multiply: n 2  10 5n  3


 xyz 1  xyz  1
2
2

ALGEBRA
p. 2
7.2 Combine and simplify algebraic expressions that contain polynomials, rational expressions, radicals, or rational exponents.
7.2 Combine and simplify algebraic expressions that contain polynomials, rational expressions, radicals, positive or negative rational exponents, and logarithmic
expressions.
High School
decides to put all of the tiles from the three desks onto
Pat’s desk. Write an algebraic expression for the new
collection of tiles on Pat’s desk
Multiply and simplify the expression
x  1x 2  2 x  3
Algebra 2
Add, subtract, multiply, or divide:
5x 2  4 x   3x 2  6 x  7
Simplify:
3
(a) 2b  3c 
(b)  5 f 12 4 f 3 g  2 f 
(c) 3x 2  2 x  4  25 x 2  7 
Multiply and simplify: x n  13 x n  3
Algebra 2/Trigonometry
Simplify:  y 3  1   y 2  1  3 y  7 
Multiply: 4x 4  x 3  bx 2  9 and x 2  2 x  3
Simplify: 2 x  1x  3  3x  3
Course not indicated
Find f  g , f  g , and f  g given f x   x  5 and
g x   x  3
Community College
EWU
ALGEBRA
p. 3
7.2 Combine and simplify algebraic expressions that contain polynomials, rational expressions, radicals, or rational exponents.
7.2 Combine and simplify algebraic expressions that contain polynomials, rational expressions, radicals, positive or negative rational exponents, and logarithmic
expressions.
High School
Community College
EWU
Core 4
Precalculus
Given the following binomials: f x   6 x 2  5 x  4 and
g x   2 x  1 find the following:
(a) f  g
f g
(c) f  g
(d) f  g
(b)
b. Factor out the greatest common factor from polynomials of any degree and from expressions involving rational exponents.
Core 1
Core 2
It is not in our curriculum
Core 3
Rewrite the following function in an equivalent
factored form:
(a) k t   7t 2  14 t
SFCC – Beginning & Intermediate Algebra
Factor:
(a) 125 a 3 z 5  60 a 4 z 4  85a 5 z 2
(b) 27 m 3  9m
SCC – Intermediate Algebra
Factor: 10 x 2 y  15 xy 2  35 xy
Basic Algebra
Which polynomials are completely factored?
(a) 5(3x – 6)
(b) x(x – 3) + 5(x – 3)
(c) 18 x 2  x
Intermediate Algebra Needs new example
Factor by factoring out the GCF:
2x  5  3x  5
(b) qx   x 3  5 x 2  6 x (made up)
Algebra Concepts
Integrated 3
Factor completely:
(a) 6x 2  30x  24
Factor ( x 2  3)
Precalculus I

2
3
 ( x 2  3)

5
3
.
ALGEBRA
p. 4
7.2 Combine and simplify algebraic expressions that contain polynomials, rational expressions, radicals, or rational exponents.
7.2 Combine and simplify algebraic expressions that contain polynomials, rational expressions, radicals, positive or negative rational exponents, and logarithmic
expressions.
High School
(b) x 4  3x 3  2x 2
(c)  5x 4  15x 3 10x 2
Similar to intermediate Algebra Example
Algebra 2, CPM
You should have noticed that each term in
12t 2 10t  2 is divisible by 2. That is, it has a
common factor of 2.
a) What is the common factor for 10x 2  25x 15 ?
b) For an expression to be completely factored, each
factor must have all common factors separated out.
Sometimes it is easiest to do this first. Since 5 is a
common factor of 10x 2  25x 15 , you can factor
10x 2  25x 15 using a special generic rectangle
which is shown below. Find the length of this generic
rectangle and write its area as a product of its length
and width.
5 10x 2
25 x
15
Factor the expression 2 x  8
Algebra 2
Factor the following expressions:
(a) 3x 3  15
(b) x 2  24x
(c) 16x 2  x
(d) xy 3  2 xy 2  xy
(e) 20 x 3  15 x 2 y 2  15 xy 3
Community College
EWU
Factor
x  17 5  x  12 5
x  12 5 x  2
ALGEBRA
p. 5
7.2 Combine and simplify algebraic expressions that contain polynomials, rational expressions, radicals, or rational exponents.
7.2 Combine and simplify algebraic expressions that contain polynomials, rational expressions, radicals, positive or negative rational exponents, and logarithmic
expressions.
High School
Community College
EWU
Algebra 2/Trigonometry
Factor out the common factor:
(a) 3xx  2  4x  2
1
(b) 4 x 3 2 x  1 2  2 x2 x  1 2
3
Course not indicated
Find the zeros of f x   x 3  x 2  6 x
Core 4
Precalculus
Factor out the GCF: 2 x 2 x  1  3xx  1
Find the zeros of the following polynomial functions:
(a) f x   3x 2  27 x
(b)
f x   3 x 3  27 x
c. Factor quadratic polynomials with integer coefficients into a product of linear terms.
Core 2
If someone claimed that t  3 is a root of the equation
0  5t 2  20t  6 how could you check that claim? Test
your method. Is t  3 a root?
Problems similar to those shown on for SFCC and
EWU
SFCC – Beginning & Intermediate Algebra
Factor:
(a) a 2  9a  20
Intermediate Algebra
Factor completely:
6x 2 10x  4 .
(b) 6 x 2  17 x  12
(c) 8 x 2  50
(d)  18 k 2  48 k  66
Algebra Concepts
Factor 25x 2  10x  1
ALGEBRA
p. 6
7.2 Combine and simplify algebraic expressions that contain polynomials, rational expressions, radicals, or rational exponents.
7.2 Combine and simplify algebraic expressions that contain polynomials, rational expressions, radicals, positive or negative rational exponents, and logarithmic
expressions.
High School
Core 3
Not very well done
Write factored forms for each of the following
quadratic function rules:
(a) f x   x 2  6 x  8
(b) g x   x 2  7 x  12
(c) hx   x 2  7 x  12
(d) ix   x 2  x  12
(e)
(f)
j x   x 2  5 x  14
k x   x 2  6 x  8
Integrated 3
Factor the following:
(a) x 2  6 x  5
(b) x 2 144
(c) 27 x 3  125 y 3
(d) 5x 2  30x  25
Similar to the Algebra Concepts example
Algebra 2, CPM
Factor the expression: x 2  5x  6
Algebra 2
Factor the following:
(a) 4x 2  16x 10
(b) 9 x 2  25
Community College
SCC – Intermediate Algebra
Factor: 12x 2  5x  2
EWU
Precalculus
Factor the following expressions:
(a) x 2  5xy  14 y 2
(b) e 2 x  3e x  2
(c) x 2n  5x n  14
(d) x n  5x  14 x  n
ALGEBRA
p. 7
7.2 Combine and simplify algebraic expressions that contain polynomials, rational expressions, radicals, or rational exponents.
7.2 Combine and simplify algebraic expressions that contain polynomials, rational expressions, radicals, positive or negative rational exponents, and logarithmic
expressions.
High School
(c)
x
2
 5 x  6
Algebra 2/Trigonometry
Factor the following trinomials:
(a)  9z 2  3z  2
(b) 12x 2  7 x  1
Factor the trinomial by graphing:
(a) 3x 2  10x  8
(b) 15x 2 11x  2
Course not indicated
Factor f x   x 2  5 x  6 and write f x  in its linear
factorization form.
Core 4
Without graphing identify all zeroes and asymptotes
for each function:
2x 2  2
(a) f x   2
x 1
x 2  x  12
(b) f x  
x4
Precalculus
Find the zeros of the polynomial functions
(a) f x   3 x 3  27 x
(b)
f x   x 2  3x  4
Community College
EWU
ALGEBRA
p. 8
7.2 Combine and simplify algebraic expressions that contain polynomials, rational expressions, radicals, or rational exponents.
7.2 Combine and simplify algebraic expressions that contain polynomials, rational expressions, radicals, positive or negative rational exponents, and logarithmic
expressions.
High School
Community College
EWU
Factor:
(a) sin 2   3 sin   2
(b) e 2 x  3e x  2
(c) x 2  3xy  2 y 2
d. Simplify quotients of polynomials given in factored form, or in a form which can be factored or determine if irreducible over the real numbers.
Core 1
Core 2
It is not in our curriculum
Core 3
Not in curriculum; do as entry task for class
Weak but…write in equivalent form (in simplest
27 p 4  9 p 2  3
possible form):
9p2
SFCC – Beginning & Intermediate Algebra
Simplify:
(a) x 2  2 x  15 x 2  6 x  5
(b)

a
2


 b 2 b  a
SCC – Intermediate Algebra
yx
Simplify:
12 x 2  12 y 2

Basic Algebra
If one of two factors of 20 x 4  10 x 3  100 x is 10x,
find the other factor using division.
Intermediate Algebra
6x  3
Reduce
to lowest terms.
2
2 x  3x  2
Algebra Concepts
y 2  7 y  18
Simplify 2
y  3y  2
Integrated 3
Simplify the following:
a 2  9a  20
a 2  25
Similar to Intermediate example
Algebra 2, CPM
Divide and simplify:
x 2  2x
4x 2

x 2  4x  4 x  2
Precalculus I
x  23  x  42
Simplify
x  4
ALGEBRA
p. 9
7.2 Combine and simplify algebraic expressions that contain polynomials, rational expressions, radicals, or rational exponents.
7.2 Combine and simplify algebraic expressions that contain polynomials, rational expressions, radicals, positive or negative rational exponents, and logarithmic
expressions.
High School
Algebra 2
Simplify:
(a) 5 3  512
(b) 200x 2  50
(c) 5w3  20w 2  3w 12
x  2x  3
(d) 
x x  2 
2
x  x6
(e)
x2  4
Algebra 2/Trigonometry
Write in simplest form:
x2 9
(a)
x 3  x 2  9x  9
y 3  2 y 2  3y
(b)
y 3 1
Precalculus
Use synthetic division to simplify the expressions:
x 3 1
(a)
x 1
x 3  x 2  x  15
(b)
x 3
Find f  g if f x  
x
and g x   x 3
x 1
Community College
EWU
ALGEBRA
p. 10
7.2 Combine and simplify algebraic expressions that contain polynomials, rational expressions, radicals, or rational exponents.
7.2 Combine and simplify algebraic expressions that contain polynomials, rational expressions, radicals, positive or negative rational exponents, and logarithmic
expressions.
High School
Use synthetic division to simplify:
Community College
EWU
x 3  x 2  x  15
x 3
e. Add, subtract, multiply, and divide two rational expressions of the form,
px 
a
where a, b, and c are real numbers such that bx  c  0 and of the form
where p(x) and
bx  c
qx 
q(x) are polynomials.
Core 1
Core 2
It is not in our curriculum
Core 3
Not in curriculum, do as entry task for class
SFCC – Beginning & Intermediate Algebra
Perform the operations and simplify.
(a) 1  1
x  3
x
(b) 1

x
* 1

x  3
 

Intermediate Algebra Too easy
Perform the operations and simplify:
2
4

(a)
x 1 x 1

(c) 3t 2  t 6t 2  15t  6t 2  t 1 2t 2  5t  25
SCC – Intermediate Algebra
Integrated 3
Add the following:
Divide as indicated:
Subtract:
5
6
x  2 2x  4
2x  6
x2 9

x 2  25 x 2  10 x  25
4
5
 2
x2 x 4
Algebra 2, CPM
Add or subtract and simplify (or multiply and divide;
see above):
2
x
3 x  18
 2
Add or subtract as indicated: 
x x  3 x  3x
(b)
2
4

x 1 x 1
Algebra Concepts
Perform the indicated operation:
4x 2  x  6
3x
5


2
x  3x  2 x  1 x  2
Precalculus I
Find the remainder when:
(a) x 5  3x 2  x  1 is divided by x 3  x
(b) x 5  3x 2  x  1 is divided by x  1
ALGEBRA
p. 11
7.2 Combine and simplify algebraic expressions that contain polynomials, rational expressions, radicals, or rational exponents.
7.2 Combine and simplify algebraic expressions that contain polynomials, rational expressions, radicals, positive or negative rational exponents, and logarithmic
expressions.
High School
2x
3x  6

x4 x4
8
4
(b)

x x2
(a)
Algebra 2
Simplify the following:
x 1
2
(a) x  2 x
1
2
x  1
x 2  5x  6
x2  4
x  2 x 1

(c)
x 1 x  2
1
1

(d)
x x2
x 2  7 x  10 x  1

(e)
x2  4
x2
(b)
Algebra 2/Trigonometry
Simplify the complex fraction:
x 
  1
2 
(a) 
x  2
Community College
EWU
ALGEBRA
p. 12
7.2 Combine and simplify algebraic expressions that contain polynomials, rational expressions, radicals, or rational exponents.
7.2 Combine and simplify algebraic expressions that contain polynomials, rational expressions, radicals, positive or negative rational exponents, and logarithmic
expressions.
High School
Community College
EWU
 x2 

x  12 

(b)
 x 

3 
 x  1 
Precalculus
Divide x 4  5x 3  6x 2  x  2 by x  2
f. Simplify products and quotients of single-term expressions with rational exponents (rationalizing denominators not necessary).
Core 2
Rewrite each of the following exponential expressions
in an equivalent form. For each, also state the general
rule that applies:
z7
(a)
z3
3
z
(b)  
t
 4 
(c)  2 
x 
(d) 3 2  3 5
1
SFCC – Beginning & Intermediate Algebra
Simplify the expression, eliminating negative
exponents. Assume all variables represent positive
numbers.
(a) 2x 2 5x 3 x 9
   
(b)
(c)
x 7
x 9
6r  2r r r 
1 2
4
Integrated 3
Simplify. Write with positive exponents:
2 3

 3 13
 u v
(b) 
2
 uv  3 w  2






4
SCC – Intermediate Algebra
1
2
Simplify: a 3  a 3
Core 3
It is not in our curriculum
5
Intermediate Algebra
Simplify the expression, eliminating negative
exponents. Assume all variables represent positive
numbers.
3
1 

(a) 9 x  2  8 xy 4 


Algebra Concepts
Consider the work below in simplifying.
 5 1
 x 4 y3
Simplify 
3
 x 4






6
ALGEBRA
p. 13
7.2 Combine and simplify algebraic expressions that contain polynomials, rational expressions, radicals, or rational exponents.
7.2 Combine and simplify algebraic expressions that contain polynomials, rational expressions, radicals, positive or negative rational exponents, and logarithmic
expressions.
High School
2 x y  3x
1
2
3
y2 
4 x 2 y 5
SFCC example
Algebra 2, CPM:
1
 1 3
 
 27 
Algebra 2
Simplify:
4
2

(a) 32 5  32 5
(b)
3x  4 x 2
2
x
3
 x4 
(c) 

 x6
3
1
3

(d) 16 x 4  2  4 xy 2




Algebra 2/Trigonometry
1
(a)
x 3  x 2
3
x 2  x 1
Community College
EWU
ALGEBRA
p. 14
7.2 Combine and simplify algebraic expressions that contain polynomials, rational expressions, radicals, or rational exponents.
7.2 Combine and simplify algebraic expressions that contain polynomials, rational expressions, radicals, positive or negative rational exponents, and logarithmic
expressions.
High School
1
(b)
Community College
EWU
5
5 2  5x 2
5 x 
3
2
Precalculus
Simplify the following expressions:
(a) x 2  x 3
3
5
2
7
(b) x 2  x 6
(c) x 3  x 3
g. Simplify products and quotients of expressions with rational exponents and rationalize denominator when necessary.
Core 1
SFCC -- Beginning & Intermediate Algebra
Simplify z
Core 2
It is not in our curriculum
1
3
z
2
3
z
1
6
z


1
6



SCC – Intermediate Algebra
Core 3
It is not in our curriculum
Integrated 3
2
1
Simplify:
x2y3z
1
3
xy 3 z 2
Easy Problems:
Precalculus I
3
Simplify
x  17 5  x  12 5
x  12 5 x  2
ALGEBRA
p. 15
7.2 Combine and simplify algebraic expressions that contain polynomials, rational expressions, radicals, or rational exponents.
7.2 Combine and simplify algebraic expressions that contain polynomials, rational expressions, radicals, positive or negative rational exponents, and logarithmic
expressions.
High School
 1
(a)  x 2 
 
(b)
Community College
EWU
4
x y 
2
1
4
Algebra 2, CPM
It is not in our curriculum
Algebra 2
Simplify:
3x
(a)
3
42 3 z
(b)
 12 32
 3 x y

2
18 x
3



2
1
y4
Algebra 2/Trigonometry
3
Simplify: 1
1
52  62
Precalculus
h. Simplify rational expressions that involve complex fractions.
Core 1
It is not in our curriculum
SFCC -- Beginning & Intermediate Algebra
Simplify:
Intermediate Algebra
ALGEBRA
p. 16
7.2 Combine and simplify algebraic expressions that contain polynomials, rational expressions, radicals, or rational exponents.
7.2 Combine and simplify algebraic expressions that contain polynomials, rational expressions, radicals, positive or negative rational exponents, and logarithmic
expressions.
High School
Core 2
It is not in our curriculum
Core 3
It is not in our curriculum
Community College
(a)
(b)
3
4
x2
2
1
x2
x 1
x 1
 x  2 2
Simplify:
x 1
x  2x  1
 x  2 2
x 1
x  2x  1
Integrated 3
x7
x 2 1
Simplify:
x 2 1
2
x  14 x  49
EWU
SCC – Intermediate Algebra
Algebra Concepts
1
If f ( x )  , find the difference quotient
x
f ( x  h)  f ( x)
, where h  0 and simplify your
h
answer.
Precalculus I
2
Easy Ones-Simplify: x
x2
5
Algebra 2, CPM
x
Simplify:
1
1
x
Algebra 2
Simplify:
Calculate the rate of change of f ( x) 
x  2 and x  a .
x  2 between
ALGEBRA
p. 17
7.2 Combine and simplify algebraic expressions that contain polynomials, rational expressions, radicals, or rational exponents.
7.2 Combine and simplify algebraic expressions that contain polynomials, rational expressions, radicals, positive or negative rational exponents, and logarithmic
expressions.
High School
(a)
3x 2  x
x 2  1
x
x  1x  1
(b)
x 2 1
x2
3x  3
x 2  5x  6
Community College
EWU
Algebra 2/Trigonometry
4  x 
2
Simplify:
1
2
 x 2 4  x 2  2
4  x2
1
Course not indicated
Confirm that f and g are inverses by showing that
x3
and
f g x   x and g  f x   x where f x  
x2
2x  3
g x  
x 1
Precalculus
f x  h   f x 
1
Simplify:
if f x  
h
x 1
i.
Simplify logarithmic expressions.
Core 2
It is not in our curriculum
SFCC -- Beginning & Intermediate Algebra
Simplify : 3 log b x  1 2 log b y  3 2 lobb z
Precalculus I
Solve: log 8 x  2  log 8 x 1  log 8 x 2  4


ALGEBRA
p. 18
7.2 Combine and simplify algebraic expressions that contain polynomials, rational expressions, radicals, or rational exponents.
7.2 Combine and simplify algebraic expressions that contain polynomials, rational expressions, radicals, positive or negative rational exponents, and logarithmic
expressions.
High School
Core 2
It is not in our curriculum
Core 3
It is not in our curriculum
Integrated 3
Simplify:
1
(a) log
10000
(b) log 4 64
(c) log 5 125
(d) ln 5.3
Algebra 2, CPM
Simplify: log 5 25
Algebra 2
Simplify: log 2 2  log 2 4  6 log 2 8
If c  log 3 10 and d  log 3 5 , express the following in
terms of c and d:
(a) log 3 50
(b) log 3 2
(c) log 3 250
(d) log 3 500
Community College
1
 0?
8
Does log 6 60  log 6 10  1?
Does log 3 8  log 3
SCC – Intermediate Algebra
EWU
 x 2 x  2 

Expand: ln 

 3x  1 
ALGEBRA
p. 19
7.2 Combine and simplify algebraic expressions that contain polynomials, rational expressions, radicals, or rational exponents.
7.2 Combine and simplify algebraic expressions that contain polynomials, rational expressions, radicals, positive or negative rational exponents, and logarithmic
expressions.
High School
Community College
EWU
Express as a single log: log 2 m  log 2 n  5
Algebra 2/Trigonometry
Condense the following expression:
1
log 4 x  1  2 log 4 x  1  6 log 4 x
2
Course not indicated
Assuming x, y, and z are positive, use properties of
logarithms to write the expression as a single
logarithm: 4 logxy   3 log yz 
Precalculus
Use properties of logarithms to write the expression as
a single logarithm:
(a) log x  log y
(b) 1 3 log x
(c) 4 log y  2 log z
Condense: 21nx  3 ln y  4 ln x  1
j. Factor polynomials over the complex numbers, if possible, and relate to the Fundamental Theorem of Algebra.
Core 2
It is not in our curriculum
SFCC -- Beginning & Intermediate Algebra
SCC – Intermediate Algebra
Core 2
It is not in our curriculum
Precalculus I
Find all its roots of the given equations and state the
multiplicity of each root.
(a) x6  16 x3  64  0
(b) x 4  6 x 3  13 x 2  24 x  36  0
ALGEBRA
p. 20
7.2 Combine and simplify algebraic expressions that contain polynomials, rational expressions, radicals, or rational exponents.
7.2 Combine and simplify algebraic expressions that contain polynomials, rational expressions, radicals, positive or negative rational exponents, and logarithmic
expressions.
High School
Core 3
It is not in our curriculum
Integrated 3
It is not in our curriculum
Algebra 2, CPM
It is not in our curriculum
Algebra 2/Trigonometry
Find all the zeros of the functions:
(a) f x   x 3  6 x 2  13 x  10
(b) g x   x 4  4 x 3  8 x 2  16 x  16
Course not indicated
Find all of the zeros and write a linear factorization of
the function. Explain how the solutions relate to the
graph: f x   6 x 4  7 x 3  x 2  67 x  105
Precalculus
Find all of the zeros and write a linear factorization of
the function f x   x 2  4 x  5
Factor f x   x 5  x 4  x 3  x 2  2 x  2 over the
rational, real and complex numbers. How many zeros
does f x  have?
Community College
EWU
ALGEBRA
p. 21
7.2 Combine and simplify algebraic expressions that contain polynomials, rational expressions, radicals, or rational exponents.
7.2 Combine and simplify algebraic expressions that contain polynomials, rational expressions, radicals, positive or negative rational exponents, and logarithmic
expressions.
High School
Find all zeros and write a linear factorization of the
function f x   x 3  4 x  5
Community College
EWU