We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

ISEE Upper Level Quantitative : Algebraic Concepts

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

Create An Account

All ISEE Upper Level Quantitative Resources

8 Diagnostic Tests 234 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #251 : Algebraic Concepts

Simplify:

$\left (x + \frac{1}{3} \right )^{3}$

Possible Answers:

$x ^{3 } + 3 x^{2} +x +\frac{1}{27}$

$x ^{3 } + x^{2} + \frac{1}{3} x +\frac{1}{27}$

$x ^{3 } + \frac{1}{3} x^{2} + \frac{1}{9} x +\frac{1}{27}$

$x ^{3 } + 2 x^{2} + \frac{2}{3} x +\frac{1}{27}$

$x ^{3 } + \frac{2}{3} x^{2} + \frac{2}{9} x +\frac{1}{27}$

Correct answer:

$x ^{3 } + x^{2} + \frac{1}{3} x +\frac{1}{27}$

Explanation:

The cube of a sum pattern can be applied here:

$\left (x + \frac{1}{3} \right )^{3}$

$= x ^{3 } + 3\cdot x^{2} \cdot \frac{1}{3}+3\cdot x \cdot \left ( \frac{1}{3} \right )^{2}+\left( \frac{1}{3} \right )^{3}$

$= x ^{3 } + x^{2} +3\cdot x \cdot \frac{1}{9} +\frac{1}{27}$

$= x ^{3 } + x^{2} + \frac{1}{3} x +\frac{1}{27}$

Report an Error

Example Question #252 : Algebraic Concepts

Fill in the box to form a perfect square trinomial:

$x ^{2} - 13x + \square$

Possible Answers:

$\frac{169}{4}$

169

$\frac{169}{2}$

Correct answer:

$\frac{169}{4}$

Explanation:

To obtain the constant term of a perfect square trinomial, divide the linear coefficient, which here is , by 2, and square the quotient. The result is

$\left (\frac{ -13}{2} \right) ^{2} = \frac{(-13)^{2}}{2^{2}}= \frac{169}{4}$

Report an Error

Example Question #251 : Algebraic Concepts

Fill in the box to form a perfect square trinomial:

$x^{2} - 18x + \square$

Possible Answers:

324

144

162

Correct answer:

Explanation:

To obtain the constant term of a perfect square trinomial, divide the linear coefficient, which here is , by 2, and square the quotient. The result is

$\left (\frac{-18}{2} \right )^{2} =\left ( -9\right ) ^{2} = 81$

Report an Error

Example Question #8 : How To Multiply Exponential Variables

Expand: $(x + 2)^{10}$

Which is the greater quantity?

(a) The coefficient of $x^{4}$

(b) The coefficient of $x^{6}$

Possible Answers:

(b) is greater.

(a) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

Correct answer:

(a) is greater.

Explanation:

Using the Binomial Theorem, if $(x + 2)^{n}$ is expanded, the $x^{r}$ term is

$_{n}\textrm{C} _{r} \cdot x^{r} \cdot 2 ^{n-r}$

$= \frac{n!}{(n-r)!r!} \cdot 2 ^{n-r} \cdot x^{r}$ .

This makes $\frac{n!}{(n-r)!r!} \cdot 2 ^{n-r}$ the coefficient of $x^{r}$ .

We compare the values of this expression at n = 10 for both r = 4 and r = 6 :

(a) $\frac{10!}{(10-4)!4!} \cdot 2 ^{10-4} = \frac{10!}{6!4!} \cdot 2 ^{6}$

(b) $\frac{10!}{(10-6)!6!} \cdot 2 ^{10-6} = \frac{10!}{4!6!} \cdot 2 ^{4} = \frac{10!}{6!4!} \cdot 2 ^{4}$

(a) is the greater quantity.

Report an Error

Example Question #9 : How To Multiply Exponential Variables

x > 3

Which is the greater quantity?

(a) $16x^{2} - 72x + 81$

(b) 8

Possible Answers:

(a) and (b) are equal

It is impossible to tell from the information given

(b) is greater

(a) is greater

Correct answer:

(a) is greater

Explanation:

$16x^{2} - 72x + 81$

$= 4^{2} x^{2} -2\cdot 4x \cdot 9 + 9^{2}$

$=\left ( 4 x \right ) ^{2} - 2 \cdot 4x \cdot 9 + 9^{2}$

$= \left ( 4x -9 \right )^{2}$

Since x > 3 ,

$4x - 9 > 4 \cdot3 - 9 = 3$ , so

$16x^{2} - 72x + 81= \left ( 4x -9 \right )^{2} > 3^{2} = 9 >8$

making (a) greater.

Report an Error

Example Question #1 : How To Multiply Exponential Variables

Which is the greater quantity?

(a) $y ^{2} + 4y + 4$

(b) $y^{2} + 6y + 9$

Possible Answers:

(a) and (b) are equal.

(b) is greater.

It is impossible to tell from the information given.

(a) is greater.

Correct answer:

It is impossible to tell from the information given.

Explanation:

We show that either polynomial can be greater by giving two cases:

Case 1: y = 0

$y ^{2} + 4y + 4 = 0 ^{2} + 4 \cdot 0 + 4 = 4$

$y^{2} + 6y + 9 = 0^{2} + 6 \cdot 0 + 9 = 9$

$y^{2} + 6y + 9 > y ^{2} + 4y + 4$

Case 2: y = -3

$y ^{2} + 4y + 4 = (-3) ^{2} + 4 \cdot (-3) + 4 = 9-12 + 4 = 1$

$y^{2} + 6y + 9 = (-3)^{2} + 6 \cdot (-3) + 9 = 9 - 18 + 9 = 0$

$y ^{2} + 4y + 4 < y^{2} + 6y + 9$

Report an Error

Example Question #931 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

and are positive integers. Which is the greater quantity?

(A) $A^{2}+B^{2}$

(B) $(A+B)^{2}$

Possible Answers:

(A) and (B) are equal

It is impossible to tell which is greater from the information given

(B) is greater

(A) is greater

Correct answer:

(B) is greater

Explanation:

$(A+B)^{2} = A^{2} + 2AB + B^{2}$

Since and are positive,

2AB > 0

$2AB +A^{2}+B^{2}> 0 +A^{2}+B^{2}$

$A^{2}+2AB +B^{2}> A^{2}+B^{2}$

$(A+B)^{2} > A^{2}+B^{2}$ for all positive and , making (B) greater.

Report an Error

Example Question #932 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

and are negative integers. Which is the greater quantity?

(A) $A^{2}+B^{2}$

(B) $(A+B)^{2}$

Possible Answers:

(A) and (B) are equal

It is impossible to tell which is greater from the information given

(B) is greater

(A) is greater

Correct answer:

(B) is greater

Explanation:

$(A+B)^{2} = A^{2} + 2AB + B^{2}$

Since and are both negative,

2AB > 0 .

$2AB +A^{2}+B^{2} > 0 +A^{2}+B^{2}$

$A^{2}+2AB +B^{2} > A^{2}+B^{2}$

$(A+B)^{2} > A^{2}+B^{2}$ for all negative and , making (B) greater.

Report an Error

Example Question #933 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

and are positive integers. Which is the greater quantity?

(A) $A^{2}-B^{2}$

(B) $(A-B)^{2}$

Possible Answers:

(A) and (B) are equal

It is impossible to tell which is greater from the information given

(A) is greater

(B) is greater

Correct answer:

It is impossible to tell which is greater from the information given

Explanation:

It is impossible to tell which is greater.

Case 1: A = 4, B= 4

Then

$A^{2}-B^{2} =4^{2}-4^{2} = 16-16 = 0$

and

$(A-B)^{2} = (4-4)^{2} = 0^{2} = 0$ .

This makes (A) and (B) equal.

Case 2: A = 5, B= 4

Then

$A^{2}-B^{2} =5^{2}-4^{2} = 25-16 = 9$

and

$(A-B)^{2} = (5-4)^{2} = 1^{2} = 1$ .

This makes (A) the greater quantity.

Report an Error

Example Question #934 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

and are positive integers greater than 1.

Which is the greater quantity?

(A) $(x+y -1)^{2}$

(B) $(x-y+1)^{2}$

Possible Answers:

(A) is greater

(A) and (B) are equal

It is impossible to determine which is greater from the information given

(B) is greater

Correct answer:

(A) is greater

Explanation:

One way to look at this problem is to substitute u = y - 1 . Since y > 1 , must be positive, and this problem is to compare $(x+u)^{2}$ and $(x-u)^{2}$ .

$(x+u)^{2} = x^{2} + 2ux + u^{2}$

and

$(x-u)^{2} = x^{2} - 2ux + u^{2}$

Since 2, , and are positive, by closure, 2ux > 0 > -2ux , and by the addition property of inequality,

$x^{2} + 2ux + u^{2} > x^{2} - 2ux + u^{2}$

$(x+u)^{2} > (x-u)^{2}$

Substituting back:

$(x+y -1)^{2} > (x-y+1)^{2}$

(A) is the greater quantity.

Report an Error

View Tutors

Gabby
Certified Tutor

University of Oregon, Bachelor in Arts, Journalism. University of Oregon, Current Grad Student, Communication, General.

View Tutors

Ana
Certified Tutor

Grand Canyon University, Bachelors, Education. Grand Canyon University, Masters, Education.

View Tutors

Crystal
Certified Tutor

SUNY at Binghamton, Bachelor of Science, Mechanical Engineering.

All ISEE Upper Level Quantitative Resources

8 Diagnostic Tests 234 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

English Tutors in Philadelphia, Chemistry Tutors in Washington DC, ACT Tutors in Dallas Fort Worth, Math Tutors in Boston, Math Tutors in Houston, GRE Tutors in Miami, Statistics Tutors in San Diego, SAT Tutors in Chicago, ACT Tutors in Seattle, Statistics Tutors in Denver

Popular Courses & Classes

GRE Courses & Classes in Atlanta, MCAT Courses & Classes in San Francisco-Bay Area, GMAT Courses & Classes in San Diego, Spanish Courses & Classes in Seattle, LSAT Courses & Classes in Phoenix, MCAT Courses & Classes in Dallas Fort Worth, LSAT Courses & Classes in Philadelphia, ISEE Courses & Classes in Philadelphia, MCAT Courses & Classes in Atlanta, SSAT Courses & Classes in Philadelphia

Popular Test Prep

ISEE Test Prep in Seattle, ISEE Test Prep in Phoenix, SSAT Test Prep in Philadelphia, LSAT Test Prep in Philadelphia, SSAT Test Prep in Miami, ISEE Test Prep in Philadelphia, SSAT Test Prep in Phoenix, SAT Test Prep in Denver, SSAT Test Prep in New York City, LSAT Test Prep in Dallas Fort Worth

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

ISEE Upper Level Quantitative : Algebraic Concepts

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #251 : Algebraic Concepts

Example Question #252 : Algebraic Concepts

Example Question #251 : Algebraic Concepts

Example Question #8 : How To Multiply Exponential Variables

Example Question #9 : How To Multiply Exponential Variables

Example Question #1 : How To Multiply Exponential Variables

Example Question #931 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Example Question #932 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Example Question #933 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Example Question #934 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

All ISEE Upper Level Quantitative Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: