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# We have a polynomial x^2 + 4x - n for an integer n between 1 and 100,

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Math Revolution GMAT Instructor
Joined: 16 Aug 2015
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GMAT 1: 760 Q51 V42
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We have a polynomial x^2 + 4x - n for an integer n between 1 and 100,  [#permalink]

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18 Feb 2020, 01:56
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Difficulty:

65% (hard)

Question Stats:

46% (02:46) correct 54% (02:21) wrong based on 26 sessions

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[GMAT math practice question]

We have a polynomial $$x^2 + 4x - n$$ for an integer $$n$$ between $$1$$ and $$100$$, inclusive. The polynomial $$x^2 + 4x - n$$ is factored into $$(x + a)(x + b)$$ where $$a$$ and $$b$$ are integers. How many different values of $$n$$ satisfy the polynomial $$x^2 + 4x – n$$?

A. $$5$$

B. $$6$$

C. $$7$$

D. $$8$$

E. $$9$$

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"Only $79 for 1 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" GMATWhiz Representative Joined: 07 May 2019 Posts: 693 Location: India Re: We have a polynomial x^2 + 4x - n for an integer n between 1 and 100, [#permalink] ### Show Tags 18 Feb 2020, 04:03 MathRevolution wrote: [GMAT math practice question] We have a polynomial $$x^2 + 4x - n$$ for an integer $$n$$ between $$1$$ and $$100$$, inclusive. The polynomial $$x^2 + 4x - n$$ is factored into $$(x + a)(x + b)$$ where $$a$$ and $$b$$ are integers. How many different values of $$n$$ satisfy the polynomial $$x^2 + 4x – n$$? A. $$5$$ B. $$6$$ C. $$7$$ D. $$8$$ E. $$9$$ Solution: • $$n$$ is an integer between $$1$$ and $$100$$. • $$x^2+4x – n$$ is factored into $$(x+a)(x+b)$$ o Thus, $$x^2+4x-n = x^2+(a+b) + ab$$  $$a + b = 4$$, and $$ab = -n$$  Either $$a$$ is negative or $$b$$ is negative. • Let’s consider $$b$$ to be negative, because that won’t change the answer because we have to deal the product of $$a$$ and $$b$$. • Let’s see how many distinct values of $$n$$ are possible. For any other value of $$a$$ or $$b$$, $$n$$ will be either less than $$1$$ or greater than $$100$$, so these are the only possible values of $$n$$. Hence, the distinct possible values of $$n$$ are $$8$$, the correct answer is Option D. _________________ GMAT Prep truly Personalized using Technology Prepare from an application driven course that serves real-time improvement modules along with a well-defined adaptive study plan. Start a free trial to experience it yourself and get access to 25 videos and 300 GMAT styled questions. Helpful Links Score Improvement Strategy: How to score Q50+ on GMAT | 5 steps to Improve your Verbal score Study Plan Articles: 3 mistakes to avoid while preparing | How to create a study plan? | The Right Order of Learning | Importance of Error Log Helpful Quant Strategies: Filling Spaces Method | Avoid Double Counting in P&C | The Art of Not Assuming anything in DS | Number Line Method Key Verbal Techniques: Plan-Goal Framework in CR | Quantifiers the tiny Game-changers | Countable vs Uncountable Nouns | Tackling Confusing Words in Main Point Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 8985 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: We have a polynomial x^2 + 4x - n for an integer n between 1 and 100, [#permalink] ### Show Tags 20 Feb 2020, 00:56 => We can assume $$a < b$$ without loss of generality. Since we have $$a + b = 4$$ and $$ab = -n$$, we have the following pairs of $$(a, b).$$ $$(-1, 5), (-2, 6), (-3, 7), (-4, 8), (-5, 9), (-6, 10), (-7, 11), (-8, 12)$$ Then, the possible values of $$n$$ are $$5, 12, 21, 32, 45, 60, 77$$ and $$96.$$ We have $$8$$ possible values of $$n$$. Therefore, the answer is D. Answer: D _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$79 for 1 month Online Course"
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Re: We have a polynomial x^2 + 4x - n for an integer n between 1 and 100,   [#permalink] 20 Feb 2020, 00:56