The expression x3 + 8 describes a mathematical function, specifically a polynomial function. To better understand its properties, we can break this down.
This expression can be rewritten as:
- x3 + 23
Recognizing this as a sum of cubes allows us to apply the sum of cubes formula, which states:
a3 + b3 = (a + b)(a2 – ab + b2)
In our case, a = x and b = 2. Applying the formula gives:
- (x + 2)(x2 – 2x + 4)
Thus, the expression x3 + 8 can also be factored as (x + 2)(x2 – 2x + 4). This tells us that:
- The function has a root at x = -2, where it intersects the x-axis.
Furthermore, the second factor, x2 – 2x + 4, is a quadratic function that opens upwards (since the coefficient of x2 is positive). The discriminant of this quadratic is:
- b2 – 4ac = (-2)2 – 4(1)(4) = 4 – 16 = -12
Since the discriminant is negative, this quadratic does not have any real roots, which means it doesn’t intersect the x-axis. Thus, it is always positive.
In summary, the function:
- x3 + 8
has one real root at x = -2, and approaches infinity as x approaches both positive and negative infinity. It is defined everywhere on the real number line and exhibits continuous behavior.