The product of three consecutive integers is 21 more than the cube of smallest integer.the smallest integer is.A.-3 B.-4 C.-5 D.-6

x^3 + 3x^2 + 2x = x^3 + 21

3x^2 + 2x - 21 = 0

(3x - 7)(x + 3) = 0

x + 3 = 0

Let's represent the three consecutive integers as x, x+1, and x+2.

According to the problem, the product of these three consecutive integers is 21 more than the cube of the smallest integer, which can be expressed as:

x * (x+1) * (x+2) = x^3 + 21

Expanding the left side of the equation, we get:

x^3 + 3x^2 + 2x = x^3 + 21

Subtracting x^3 from both sides of the equation, we have:

3x^2 + 2x = 21

Rearranging the equation:

3x^2 + 2x - 21 = 0

Now we can solve this quadratic equation by factoring or using the quadratic formula.

Factoring method:
(3x + 7)(x - 3) = 0

Setting each factor to zero:
3x + 7 = 0 or x - 3 = 0

Solving these equations, we find:
3x = -7
x = -7/3 (Approximately -2.33)

or

x = 3

Since we are looking for the smallest integer, the correct answer is A. -3.

To solve this problem, let's assume the smallest integer is represented by "x".

According to the problem statement, the product of three consecutive integers can be expressed as:
x * (x + 1) * (x + 2)

And it is given that this product is 21 more than the cube of the smallest integer, which can be expressed as:
x^3 + 21

So, we have the equation:
x * (x + 1) * (x + 2) = x^3 + 21

Next, we can simplify this equation and solve for "x".

Expanding the left side:
x * (x + 1) * (x + 2) = x^3 + 3x^2 + 2x

Now, let's bring everything to one side of the equation:
x^3 + 3x^2 + 2x - (x^3 + 21) = 0

Simplifying further:
2x^2 + 2x - 21 = 0

To solve this quadratic equation, we can either factorize it or use the quadratic formula. But since the options provided are negative integers, let's try the values one by one.

Let's start with option C, which is -5.

When x = -5:
2(-5)^2 + 2(-5) - 21 = 0
50 - 10 - 21 = 0
39 - 21 = 0
18 ≠ 0

Option C does not satisfy the equation.
Let's move to the next option, which is D, -6.

When x = -6:
2(-6)^2 + 2(-6) - 21 = 0
2(36) - 12 - 21 = 0
72 - 12 - 21 = 0
60 - 21 = 0
39 ≠ 0

Option D also does not satisfy the equation.

Let's try the remaining options.

When x = -3:
2(-3)^2 + 2(-3) - 21 = 0
2(9) - 6 - 21 = 0
18 - 6 - 21 = 0
12 - 21 = 0
-9 = 0

Option A satisfies the equation.

Therefore, the smallest integer is -3.

So, the answer is A. -3.