If k - x is a factor of the expression - x 2 + 1 29 n k 2 ,

Question

If k - x is a factor of the expression - x 2 + 1 29 n k 2 , where n and k are constants and k &#62; 0 , what is the value of n ?

Accepted Answer

Choice D is correct. If k - x is a factor of the expression - x 2 + 1 29 n k 2 , then the expression can be written as k - x a x + b , where a and b are constants. This expression can be rewritten as a k x + b k - a x 2 - b x , or - a x 2 + a k - b x + b k . Since this expression is equivalent to - x 2 + 1 29 n k 2 , it follows that - a = - 1 , a k - b = 0 , and b k = 1 29 n k 2 . Dividing each side of the equation - a = - 1 by - 1 yields a = 1 . Substituting 1 for a in the equation a k - b = 0 yields k - b = 0 . Adding b to each side of this equation yields k = b . Substituting k for b in the equation b k = 1 29 n k 2 yields k 2 = 1 29 n k 2 . Since k is positive, dividing each side of this equation by k 2 yields 1 = 1 29 n . Multiplying each side of this equation by 29 yields 29 = n . Alternate approach: The expression x 2 - y 2 can be written as x - y x + y , which is a difference of two squares. It follows that 1 29 n k 2 - x 2 is equivalent to 1 29 n k - x 1 29 n k + x . It&rsquo;s given that k - x is a factor of - x 2 + 1 29 n k 2 , so the factor 1 29 n k - x is equal to k - x . Adding x to both sides of the equation 1 29 n k - x = k - x yields 1 29 n k = k . Since k is positive, dividing both sides of this equation by k yields 1 29 n = 1 . Squaring both sides of this equation yields 1 29 n = 1 . Multiplying both sides of this equation by 29 yields n = 29 . Choice A is incorrect. This value of n gives the expression - x 2 + 1 29 - 29 k 2 , or - x 2 - k 2 . This expression doesn't have k - x as a factor. Choice B is incorrect. This value of n gives the expression - x 2 + 1 29 - 1 29 k 2 , or - x 2 + - 1 841 k 2 . This expression doesn't have k - x as a factor. Choice C is incorrect. This value of n gives the expression - x 2 + 1 29 1 29 k 2 , or - x 2 + 1 841 k 2 . This expression doesn't have k - x as a factor.

Question if-k-x