Keystone Remediation Packet Week 1
Operations with Real Numbers and Expressions
Assigned: Friday, April 5
Due: Friday, April 12
You must show all work to receive full credit
The order of operations mandates that you perform parenthesis, exponents, multiplication and division, and then
addition and subtract in that order. Real numbers are all positive and negative numbers that either terminate or repeat.
1] Which of the following is true for all real values of x?
A. x3  x 2
B. 3 x 2  2 x 3
C.  2 x   3x 2
2
D. 3  x  2   3 x 2  2
2
Perfect Squares are the product of an integer times itself. Taking the square root of a perfect square results in the
original integer.
2] Which value of x makes the expression 2 51x equivalent to 10 51 ?
A. 5
B. 25
C. 50
D. 100
Factors are numbers that multiply to make a product. If a factor is repeated it forms a perfect square factor which can be
easily removed from the radical.
3] For which value of x should the expression
A. x  10
B. x  13
C. x  21
D. x  38
87x be further simplified?
A least common multiple of two numbers can be found by listing multiples of each number and finding the smallest in
common. Furthermore, variable expressions can be included in least common multiples by choosing the highest power of
every variable represented in both terms.
4] What is the least common multiple of the two monomials that follow? 450 x 2 y 5 & 3,000 x 4 y 3
A. 2xy
B. 30xy
C. 150x 2 y 3
D. 9, 000x 4 y 5
Negative exponents move their base from a numerator to the denominator or from a denominator to the numerator.
Once the base has been moved, the exponent’s sign changes to a positive but the number itself does not change.

5] Simplify the following expression: 2 2 4

2
1
8
1
B.
4
A.
C. 16
D. 32
When estimating, try to round given numbers to multiples of 10. Then manipulate those rounded numbers to create an
estimate of what is being asked for.
6] A theme park charges $52 for a day pass and $110 for a week pass. Last month, 4,432 day passes were sold and 979
week passes were sold. Which is the closest estimate of the total amount of money paid for the day and week passes for
last month?
A. $300, 000
B. $400, 000
C. $500, 000
D. $600, 000
When multiplying polynomials, distribute every term in the first polynomial to every term in the second polynomial using
multiplication. Then, simplify by combining like terms that result.
7] The polynomial expression (mx3  3)(2 x 2  5x  2)  (8x5  20 x 4 ) can be simplified to 8 x 3  6 x 2  15 x  6 . What is
the value of m?
A. 8
B. 4
C. 4
D. 8
Factoring is the process of removing a common factor from many terms using division. If a factor is common in the
numerator and denominator of an expression it will simplify to 1.
8] Simplify the expression
1 2 5
x  x
2
4
1
5
B. x 3  x 2  x
2
4
x5
C.
x4
x5
D.
x4
A.
3x3  9 x 2  30 x
given the restrictions x  4, 2, 0 .
3x3  18 x 2  24 x
To factor a quadratic trinomial in standard form (ax2+bx+c), start by finding factors of c that add to make b.
9] What is a factor of x 2  3x  18 , when it is factored completely.
A. ( x  2)
B. ( x  3)
C. ( x  6)
D. ( x  9)
When writing an equation, a onetime fee or cost is expressed using addition or subtraction. A fee that increases or
decreases with another amount is expressed using multiplication or division. To see when equations will have the same
solution, set one equation equal to the other and solve for the remaining variable.
10] Sam needs to make a long-distance call from a pay phone. With his prepaid phone card, he will be charged $1.00 to
connect and $0.50 per minute. If he places a collect call with the operator he will be charged $3.00 to connect and $0.25
per minute. In how many minutes will the phone card and the collect call cost the same?
A. 5min
B. 5 13 min
C. 8min
D. 16min
To factor a quadratic trinomial in standard form (ax2+bx+c) where a≠1, start by finding factors of the product a∙c that
add to make b. Rewrite the linear term as two terms using the factors you just identified, and then factor by grouping.
11] When the expression 2𝑥 2 + 13𝑥 − 24 is factored completely, which is one of its factors?
A. (2 x  4)
B. ( x  6)
C. (2 x  3)
D. ( x  8)
When you subtract one polynomial from another, be sure to distribute negative one to all terms of the second polynomial
before removing parenthesis and combining like terms.
12] What is the difference when 3𝑔2 − 4𝑔 + 2 is subtracted from 7𝑔2 + 5𝑔 − 12?
A. 4 g 2  9 g  3
B. 4 g 2  g  1
C. 4 g 2  9 g  14
D. 10 g 2  g  1
When multiplying common bases, add the bases’ exponents to determine the exponent of the product.
13] What is the product of (−3𝑥 2 𝑦) and(5𝑥𝑦 2 + 𝑥𝑦)?
A. 15 x3 y 3  3x3 y 2
B. 15 x3 y 3  3x3 y
C. 15 x 2 y 2  3x 2 y
D. 15x3 y 3  xy
To find when an equation will generate a given amount, set the equation equal to that amount and solve for the
remaining variable.
14] The cost to rent a construction crane is $750 per day plus $250 per hour for use. What is the maximum number of
hours the crane can be used each day if the rental cost is not to exceed $2500 per day?
A. 2.5
B. 3.7
C. 7.0
D. 13.0
To compare numbers, change all of the expressions to their decimal form. To convert a percentage to a decimal, divide
the percentage by 100. The direction in which you read an inequality determines what kind of inequality it is. For example
5<6 can be read as, “five is less than six” or as, “six is greater than five.”
15] Choose the correct comparison for the following numbers: 0.46,104%,
5
 104%
14
5
 0.46
B. 104% 
14
5
C. 104%  0.46 
14
5
 104%  0.46
D.
14
A. 0.46 
5
14