GMAT Math : Solving by Factoring

Study concepts, example questions & explanations for GMAT Math

Create An Account

All GMAT Math Resources

22 Diagnostic Tests 693 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Solving By Factoring

If and are positive, what is the value of x-y ?

(1) $(x )^{2}-(y )^{2}=15$

(2) x+y=5

Possible Answers:

B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

D. EACH statement ALONE is sufficient.

E. Statements (1) and (2) TOGETHER are NOT sufficient.

Correct answer:

C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Explanation:

For statement (1), we can factor the equation as following: $(x )^{2}-(y )^{2}=(x+y)*(x-y)$ . Obviously we cannot figure out x-y since we have no information about the value of x+y .

From statement (2) only, we have no idea what x-y is by knowing the value of x+y . However, putting the two statements together, we will get x-y=3 .

Report an Error

Example Question #2 : Dsq: Understanding Factoring

Don's teacher challenged him to write two whole numbers in the square and circle in the diagram below in order to make a polynomial that could be factored.

$x^{2}+ \square x+ \bigcirc$

Assuming that Don wrote two whole numbers, did he succeed?

Statement 1: The number Don wrote in the circle is the square of half the number he wrote in the square.

Statement 2: The sum of the numbers Don wrote in the square and in the circle is 15.

Possible Answers:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Correct answer:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Explanation:

By Statement 1 alone, if we call the number in the square , then the number in the circle is the square of half this, or $A^{2}$ ; the polynomial is therefore

$x^{2}+ 2A x+ A^{2}$ ,

which is the pattern of a perfect square trinomial. The polynomial can be factored, and Don succeeded.

Assume Statement 2 alone. The polynomials

$x^{2}+6 x+9$

and

$x^{2}+ 5x+ 10$

fit the conditions of Statement 2.

Some trinomial of the pattern $x^{2}+ Bx+ C}$ can be factored as (x+M)(x+N) , where and have product and sum .

$x^{2}+6 x+9 = (x+3)(x+3)$ , since 3 and 3 have sum 6 and product 9.

However,

$x^{2}+ 5x+ 10$ cannot be factored; no two integers exist with product 10 and sum 5.

Report an Error

Example Question #1 : Solving By Factoring

Karen's teacher challenged her to write two whole numbers in the square and circle in the diagram below in order to make a polynomial that could be factored.

$\square x ^{3}+ \bigcirc y^{3}$

Assuming both numbers are whole numbers, did Karen succeed?

Statement 1: The cube root of the number Julia wrote in the circle is a whole number.

Statement 2: The number Julia wrote in the square is twenty-seven times the number Julia wrote in the circle.

Possible Answers:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Correct answer:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Explanation:

A binomial of the form $Ax^{3}+By^{3}$ can be factored out as the sum of cubes if and only if both and are perfect cubes, and it can be factored out by taking out a GCF if the GCF of and is not 1.

Statement 1 alone gives no clue as to the number that was placed in the square. The polynomials $11x^{3} + 8y^{3}$ and $12x^{3} + 8y^{3}$ fit the statement, but neither can be factored as the sum of cubes, and only the latter can have a greatest common factor (4) factored out.

Assume Statement 2 alone. There are two possible cases:

Case 1: Julia wrote a 1 in the circle.

The polynomial is $27 x ^{3}+1 y^{3} = 27 x ^{3}+ y^{3} = \left (3x \right ) ^{3}+ y^{3}$ , which, as the sum of cubes, is factorable.

Case 2: Julia wrote a different whole number in the circle.

Since the number in the square is 27 times the number in the circle, at the very least, the polynomial can be factored by distributing out the number in the circle. For example,

$54x ^{3}+2y^{3} = 2 (27x^{3}+y^{3})$

Report an Error

Example Question #4 : Dsq: Understanding Factoring

Julia's teacher challenged her to write two whole numbers in the square and circle in the diagram below in order to make a polynomial that could be factored.

$\square x ^{3}+ \bigcirc y^{3}$

Did Julia succeed?

Statement 1: The cube root of the number Julia wrote in the circle is a whole number.

Statement 2: The number Julia wrote in the square is ten times the number Julia wrote in the circle.

Possible Answers:

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

A binomial of the form $Ax^{3}+By^{3}$ can be factored out as the sum of cubes if and only if both and are perfect cubes, and it can be factored out by taking out a GCF if the GCF of and is not 1.

Assume both statements to be true.

$10x ^{3}+1y^{3}$ - which is equal to $10x ^{3}+ y^{3}$ - fits the conditions. But the polynomial cannot be factored using the sum of cubes property (10 is not a cube, but 1 is), nor can a greatest common factor be taken out (the greatest common factor of the terms is 1).

$80x ^{3}+8y^{3}$ fits the conditions, and can be factored out as

$80x ^{3}+8y^{3} = 8 \cdot 10 x^{3} +8 \cdot y^{3}= 8 \left (10 x^{3} +8 y^{3} \right )$

Report an Error

Example Question #5 : Dsq: Understanding Factoring

Chad's teacher challenged him to write two whole numbers in the square and circle in the diagram below in order to make a polynomial that could be factored.

$\square x ^{3} - \bigcirc y^{3}$

Did Chad succeed?

Statement 1: The cube root of the number Chad wrote in the square is a whole number.

Statement 2: The cube root of the number Chad wrote in the circle is an irrational number.

Possible Answers:

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

Assume both statements are true. To try to factor the polynomial, we have one of two possibilities - the difference of cubes, or a greatest common factor. But since the number in the circle is not a perfect cube, this leaves us with factoring by GCF.

The polynomial $27x^{3} - 7y^{3}$ fits both statements; GCF ( 27,7)= 1 , so it cannot be factored.

The polynomial $27x^{3} - 9y^{3}$ fits both statements; GCF ( 27,9)=9 , so it can be factored:

$27x^{3} - 9y^{3} = 9(3x^{3} - y^{3} )$

Report an Error

Example Question #6 : Dsq: Understanding Factoring

Willy's teacher challenged him to write two whole numbers in the square and circle in the diagram below in order to make a polynomial that could be factored.

$\square x ^{2} - \bigcirc$

Did Willy succeed?

Statement 1: The number Willy wrote in the square is a multple of 4.

Statement 2: The number Willy wrote in the circle is a multiple of 9.

Possible Answers:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

Assume both statements are true.

If Willy wrote a 4 in the circle and a 9 in the square, both statements are satisfied, and the resulting polynomial can be factored as the difference of squares:

$4 x ^{2} - 9 = (2x)^{2} - 3^{3} = (2x+3)(2x-3)$

If Willy wrote a 4 in the circle and a 27 in the square, both statements are satisfied. However, the resulting polynomial

$4 x ^{2} - 27$

is prime; no greatest common factor can be taken out, and the only possible pattern, the difference of squares, does not fit, since the square root of 27 is irrational.

The two statements together are insufficient.

Report an Error

Example Question #7 : Dsq: Understanding Factoring

Theresa's teacher challenged her to write whole numbers in the circle and the square in the diagram below in order to make a polynomial that could be factored.

$x^{2} - \bigcirc y ^{\square}$

Assuming Theresa wrote two whole numbers, did she succeed?

Statement 1: Theresa wrote a 64 in the circle.

Statement 2: Theresa wrote a multiple of 6 in the square.

Possible Answers:

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

The template fits the difference of squares pattern, which may be used only if all coefficients are perfect squares and all exponents are even (making the powers of the variables perfect squares). Each statement only answers the question of one of these qualifications; the two together answer both.

Report an Error

Example Question #8 : Dsq: Understanding Factoring

Consider function f(x) .

I) f(x) has zeroes at x=4 and x=5 .

II) f(x) is a second degree polynomial.

Find the equation that models f(x) .

Possible Answers:

Either statement alone is sufficient to answer the question.

Statement II is sufficient to answer the question, but statement I is not sufficient to answer the question.

Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.

Both statements are necessary to answer the question.

Neither I nor II are sufficient to answer the question. More information is needed.

Correct answer:

Neither I nor II are sufficient to answer the question. More information is needed.

Explanation:

If we consider statement II, we know f(x) must be of the form

$\small y=Ax^2+Bx+C$ .

Due to the zero product property and statement I, we know that f(x) CAN look like this

$\small f(x)=(x-4)(x-5)$ .

However, it could also look like this

$\small \small f(x)=9(x-4)(x-5)$ .

Or this

$\small \small f(x)=-1500(x-4)(x-5)$ .

So with what we are given it is impossible to find the true equation for f(x).

Report an Error

Example Question #9 : Dsq: Understanding Factoring

Simplify: $\frac{-3x + 3}{x^2 - 1}$

Possible Answers:

$\frac{-3}{x-1}$

$\frac{-3x}{x-1}$

$\frac{3}{x-1}$

$\frac{-3}{x+1}$

$\frac{-3x}{x+1}$

Correct answer:

$\frac{-3}{x+1}$

Explanation:

In order to simplify, we must first pull out the largest common factor of each term in the numerator, -3:

$\frac{-3x+3}{x^2 - 1} = \frac{-3(x-1)}{x^2-1}$

We then recognize that the denominator is a difference of squares:

$\frac{-3(x-1)}{x^2-1} = \frac{-3(x-1)}{(x-1)(x+1)}$

We can therefore cancel the (x-1) terms and are left with:

$\frac{-3}{x+1}$

Report an Error

All GMAT Math Resources

22 Diagnostic Tests 693 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

SSAT Tutors in Atlanta, Physics Tutors in Dallas Fort Worth, Algebra Tutors in Atlanta, Statistics Tutors in Washington DC, LSAT Tutors in Atlanta, Physics Tutors in Seattle, ACT Tutors in San Francisco-Bay Area, Algebra Tutors in Denver, Spanish Tutors in Chicago, GMAT Tutors in Denver

Popular Courses & Classes

ISEE Courses & Classes in Atlanta, SSAT Courses & Classes in Miami, Spanish Courses & Classes in New York City, GMAT Courses & Classes in Washington DC, Spanish Courses & Classes in San Francisco-Bay Area, SSAT Courses & Classes in Dallas Fort Worth, Spanish Courses & Classes in Phoenix, ACT Courses & Classes in Boston, GRE Courses & Classes in Chicago, LSAT Courses & Classes in Dallas Fort Worth

Popular Test Prep

ACT Test Prep in Philadelphia, GMAT Test Prep in New York City, ISEE Test Prep in Phoenix, LSAT Test Prep in Miami, LSAT Test Prep in Dallas Fort Worth, SAT Test Prep in Dallas Fort Worth, ACT Test Prep in Boston, SSAT Test Prep in Miami, GRE Test Prep in Chicago, MCAT Test Prep in Los Angeles

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)

Email address:
Your name:
Feedback:

GMAT Math : Solving by Factoring

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #1 : Solving By Factoring

Example Question #2 : Dsq: Understanding Factoring

Example Question #1 : Solving By Factoring

Example Question #4 : Dsq: Understanding Factoring

Example Question #5 : Dsq: Understanding Factoring

Example Question #6 : Dsq: Understanding Factoring

Example Question #7 : Dsq: Understanding Factoring

Example Question #8 : Dsq: Understanding Factoring

Example Question #9 : Dsq: Understanding Factoring

All GMAT Math Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details