Math question....?

Question

What are all the positive integral values of x for which the expression (x^3-12)/ x-4 has an integral value?

What does integral mean?

Answer 1

integers are whole numbers positive and negative including zero
for x=0 x^3-12/x-4=3
forx=all even numbers like 2,6 etc except x=4

Answer 2

What are all the positive integral values of x for which the expression (x^3-12)/ x-4 has an integral value?

(x³ - 12)/ (x - 4)

= (x³ - 64 + 52)/(x - 4)

= (x³ - 4³)/(x - 4) + 52/(x - 4)

= x² + 4x + 16 + 52/(x - 4) (x ≠4)

So
∫[(x³ - 12)/ (x - 4)]dx

= ∫[x² + 4x + 16 + 52/(x - 4)]dx (x ≠4)

= ⅓x³ + 2x² + 16x + 52 ln|x - 4| + C (x ≠4)

ie integral value exists for all x>0 except x = 4 (positive x being your restriction. In fact it exists for all x except x = 4

What does integral mean?

The integral is the 'antiderivative' ... the function that needs to be differentiated to get that derivative given.

In the example above

d/dx(⅓x³ + 2x² + 16x + 52 ln|x - 4| + C) (x ≠4)
= x² + 4x + 16 + 52/(x - 4) (x ≠4)
= (x³ - 12)/ (x - 4) (x ≠4)

Alternate definition .. do you mean integer values??

ie x can only have values that are positive whole numbers (positive integers) and the expression values can only be whole numbers (integers)

If so then

(x³ - 12)/ (x - 4)

= (x³ - 64 + 52)/(x - 4)

= (x³ - 4³)/(x - 4) + 52/(x - 4)

= x² + 4x + 16 + 52/(x - 4) (x ≠4)

Firstly:

x ≠4

Secondly:

x² + 4x + 16 is always an integer when x is an integer so (x - 4) must also be a factor of 52 for x² + 4x + 16 + 52/(x - 4) to be an integer

If x > 0 (your definition) then

x - 4 = ..... .......... -2 ... -1 ..... 1 ..... 2 ...... 4 ..... 13 .... 26 ..... 52
x = ........... ........... 2 .... 3 ..... 5 ..... 6 ....... 8 ..... 17 .... 30 ..... 56
(x³ - 12)/ (x - 4) = 2 .. -15 . 113 . 102 . 125 . 377 . 1038 . 3377

Answer 3

"integral" means "is an integer", which means 0, 1, -1, and so on. Positive means you only care about 1, 2, 3...

So any fraction is integral when the denominator divides the numerator. This means you need to find integers such that x - 4 | x^3 - 12.

This isn't so easy as it is - try letting y = x - 4, x = y + 4. Then the equation becomes

y | (y + 4)^3 - 12

y | y^3 + 12y^2 + 48y + 64 - 12 = y^3 + 12y^2 + 48y + 52

y clearly divides the first 3 terms, so the complete answer is the integers that divide 52. These are y = +-(1, 2, 4, 13, 26, 52). You only care about positive x, which means y > -4, so the answers are
y = -2, -1, 1, 2, 4, 13, 26, 52, x = 2, 3, 5, 6, 17, 30, 56. (Unless I made a calculation error.)
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No, in this context "integral" does not refer to the calculus antiderivative. This context ("positive integral values) clearly means integers.

Answer 4

Most likely, the word is "integer".
"Integral" is a a term that means "including everything", and, in maths, means the sum of all values between two limits (or between -inf and +inf.