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Question 29 of 30

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M:C:MC:22

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SAT

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If the expression fraction numerator 4 x squared over denominator 2 x minus 1 end fraction is written in the equivalent form fraction numerator 1 over denominator 2 x minus 1 end fraction plus A comma what is A in terms of x?

Select an Answer

2 x plus 1

Correct Answer: 
Yes

2 x minus 1

Correct Answer: 
No

4 x squared

Correct Answer: 
No

4 x squared minus 1

Correct Answer: 
No

View Correct Answer

Choice A is correct. The form of the equation suggests performing long division on fraction numerator 4 x squared over denominator 2 x minus 1 end fraction :

space space space space space space space space space space space space space 2 x plus 1
2 x minus 1 long division enclose 4 x to the power of 2 space space space space space space space space space space space space space space space space space space end exponent end enclose
space space space space space space space space space space space space bottom enclose 4 x to the power of 2 space end exponent minus 2 x end enclose
space space space space space space space space space space space space space space space space space space space space space space space 2 x
space space space space space space space space space space space space space space space space space space space space space space space bottom enclose 2 x minus 1 end enclose
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space 1

Since the remainder 1 matches the numerator in fraction numerator 1 over denominator 2 x minus 1 end fraction comma it is clear that A equals 2 x plus 1.

A short way to find the answer is to use the structure to rewrite the numerator of the expression as left parenthesis 4 x squared minus 1 right parenthesis plus 1 comma recognizing the term in parentheses as a difference of squares, making the expression equal to fraction numerator left parenthesis 2 x minus 1 right parenthesis left parenthesis 2 x plus 1 right parenthesis plus 1 over denominator 2 x minus 1 end fraction equals 2 x plus 1 plus fraction numerator 1 over denominator 2 x minus 1 end fraction. From this, the answer 2 x plus 1 is apparent. Another way to find the answer is to isolate A in the form A equals fraction numerator 4 x squared over denominator 2 x minus 1 end fraction minus fraction numerator 1 over denominator 2 x minus 1 end fraction and simplify. As with the first approach, this approach also requires students to recognize 4 x squared minus 1 as a difference of squares that factors.

Choice B is not the correct answer. The student may have made a sign error while subtracting partial quotients in the long division.

Choice C is not the correct answer. The student may misunderstand how to work with fractions and may have tried the incorrect calculation
fraction numerator 4 x squared over denominator 2 x minus 1 end fraction equals fraction numerator left parenthesis 1 right parenthesis left parenthesis 4 x squared right parenthesis over denominator 2 x minus 1 end fraction equals fraction numerator 1 over denominator 2 x minus 1 end fraction plus 4 x squared.

Choice D is not the correct answer. The student may misunderstand how to work with fractions and may have tried the incorrect calculation
fraction numerator 4 x squared over denominator 2 x minus 1 end fraction equals fraction numerator 1 plus 4 x squared minus 1 over denominator 2 x minus 1 end fraction equals fraction numerator 1 over denominator 2 x minus 1 end fraction plus 4 x squared minus 1.

Question Difficulty: 
hard
Objective: 

Students must transform a given expression into a more useful form (from improper to proper rational form).