Example of Cubic Function

Introduction of cubic function:

A Cubic function is a small different from a quadratic function. A Cubic functions have a 3 x intercept, The cubic function refer to as 3 degrees. The example of a cubic function is y=(x-1)(x+3)(x-4). it has 3 x intercepts which loaded on (1,0)(-3,0)(4,0).A cubic function is one of the functions which is formed as, F(x) =ax3+bx2+cx+dwhere a- nonzero (or) say polinomial of degree three. Quadratic function is derivation for cubic function. Also, a intergral for a cubic function.By ƒ(x) = 0 and assuming a ≠ 0 gives the cubic formula of the form:ax3+bx2+cx+d=0Coefficient a, b, c, d are real numbers. However, most of theory is also legal if they belong to field of characteristic other than 2 or 3.

Example Problems on Cubic Function:

Roots of a cubic function:

Each cubic equation with real coefficients have at least one solution x among the real numbers; this is a consequence of the Intermediate value theorem. We are able to differentiate several likely cases using the discriminant.

`Delta=18abcd-4b^3d+b^2c^2-4ac^3-36a^2d^2`

The next cases require to be measured

If Δ > 0, the equation have three distinct real roots.
If Δ = 0, the equation has a multiple root along with all its roots are real.
If Δ < 0, t the equation have one real root along with two non real complex conjugate roots.


Let us see some examples of cubic function:
Example 1:

Solving the factors of the cubic of the equation  x3-3x2-25x+75.

Solution:

The given equation is x3-3x2-25x+75.

This form as ax3 + bx2 + cx + d

So,  (x3-3x2) + (-25x+75)

Take a common variable:

=x2(x-3) -25(x-3)

=( x2-25) (x-3)

Here x2 – 25 in the form of a2 + b2 = (a + b) (a - b)

So, x2 – 25 = (x + 5)(x - 5)

=(x-5)(x+5)(x-3)

Answer: The solutions are 5,-5,3

Example 2:

Solving the factors of the cubic of the equation 6x3-36x2 = -54x

Solution:

This equation can be written as 6x3-36x2 + 54x=0.

This form as ax3 + bx2 + cx + d

So, 6x(x2-6x+9) =0.

Here x2 – 6x + 9 in the form of Ax2 + Bx + C so we find the factor.I like to share this Free math problem solver with you all through my article.

6x(x-3)(x-3) =0.

x=0, x=3, x=3.

Answer: The solutions are 0, 3, and 3.

Example 3:

Solving the factors of the cubic  of the equation x3-4x2-100x+400.

Solution:

The given cubic equation is (x3-4x2) + (-100x+400)

=x2(x-4) -100(x-4)

=( x2-100) (x-4)

=(x-10)(x+5)(x-3)

Answer : The solutins are 5,-5,3