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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8985
GMAT 1: 760 Q51 V42
GPA: 3.82
We have a polynomial x^2 + 4x - n for an integer n between 1 and 100,  [#permalink]

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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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GMATWhiz Representative G
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]

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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.
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Math Revolution GMAT Instructor V
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]

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=>

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 $$n$$.

_________________ Re: We have a polynomial x^2 + 4x - n for an integer n between 1 and 100,   [#permalink] 20 Feb 2020, 00:56

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