BackClose

Select a Subject

Subject Test Math 2

Question 3 of 28

beginning of content:
Tags:
SAT Subject Test

If x minus 2 is a factor of x to the power of 3 end exponent plus k x to the power of 2 end exponent plus 12 x minus 8, then k equals

Select an Answer

negative 6

Correct Answer: 
Yes

negative 3

Correct Answer: 
No

2

Correct Answer: 
No

3

Correct Answer: 
No

6

Correct Answer: 
No

The correct answer is A.

By the Factor Theorem, x minus 2 is a factor of x to the power of 3 end exponent plus k x to the power of 2 end exponent plus 12 x minus 8 only when 2 is a root of x to the power of 3 end exponent plus k x to the power of 2 end exponent plus 12 x minus 8 comma that is, left parenthesis 2 right parenthesis to the power of 3 end exponent plus k left parenthesis 2 right parenthesis to the power of 2 end exponent plus 12 left parenthesis 2 right parenthesis minus 8 equals 0, which simplifies to 4 k plus 24 equals 0. Therefore, k equals negative 6.

Alternatively, one can perform the division of x to the power of 3 end exponent plus k x to the power of 2 end exponent plus 12 x minus 8 by x minus 2 and then find a value for k so that the remainder of the division is 0.

x to the power of 3 end exponent plus k x to the power of 2 end exponent plus 12 x minus 8 equals x to the power of 2 end exponent left parenthesis x minus 2 right parenthesis plus left parenthesis 2 plus k right parenthesis x to the power of 2 end exponent plus 12 x minus 8

text    end text text     end text equals x squared left parenthesis x minus 2 right parenthesis plus left parenthesis 2 plus k right parenthesis x left parenthesis x minus 2 right parenthesis plus left parenthesis 2 left parenthesis 2 plus k right parenthesis plus 12 right parenthesis x minus 8

text    end text text     end text equals x to the power of 2 end exponent left parenthesis x minus 2 right parenthesis plus left parenthesis 2 plus k right parenthesis x left parenthesis x minus 2 right parenthesis plus left parenthesis 2 left parenthesis 2 plus k right parenthesis plus 12 right parenthesis left parenthesis x minus 2 right parenthesis plus left parenthesis 2 left parenthesis 2 left parenthesis 2 plus k right parenthesis plus 12 right parenthesis minus 8 right parenthesis

text    end text text     end text equals left parenthesis x to the power of 2 end exponent plus left parenthesis 2 plus k right parenthesis x plus left parenthesis 2 left parenthesis 2 plus k right parenthesis plus 12 right parenthesis right parenthesis left parenthesis x minus 2 right parenthesis plus left parenthesis 8 plus 4 k plus 24 minus 8 right parenthesis

text    end text text     end text equals left parenthesis x to the power of 2 end exponent plus left parenthesis 2 plus k right parenthesis x plus left parenthesis 2 left parenthesis 2 plus k right parenthesis plus 12 right parenthesis right parenthesis left parenthesis x minus 2 right parenthesis plus left parenthesis 4 k plus 24 right parenthesis

Since the remainder is 4 k plus 24, the value of k must satisfy 4 k plus 24 equals 0. Therefore, k equals negative 6.

Question Difficulty: 
easy