Answer:

(a - 6)^3

Step-by-step explanation:

Given,

a^3 - 18a^2 + 108a - 216

F(a) = a^3-18a^2+108a-216

Polynomial Roots Calculator is a set of methods aimed at FINDING VALUES of a for which F(a)=0

Rational Roots Test is ONE of the above mentioned tools. It would only find Rational Roots that is numbers a which can be expressed as the quotient of two integers

The Rational Root Theorem states that if a polynomial zeroes for a rational number P/Q then P is a factor of the Trailing Constant and Q is a factor of the Leading Coefficient

In this case, the Leading Coefficient is 1 and the Trailing Constant is -216.

The factor(s) are:

of the Leading Coefficient : 1

of the Trailing Constant : 1 ,2 ,3 ,4 ,6 ,8 ,9 ,12 ,18 ,24 , etc.

In our case this means that

a^3-18a^2+108a-216

can be divided with a-6 .

On dividing a^3-18a^2+108a-216 by (a-6), we get a^2 - 12a + 36.

So now,

p(X) = a^2 - 12a + 36.

On splitting the middle TERM, we get,

a^2 - 6a - 6a + 36 = 0

=> a(a - 6) -6(a - 6) = 0

=> (a - 6)(a - 6) = 0

Here, either (a - 6) or (a - 6) is equal to zero.

=> (a - 6) = 0 or (a - 6) = 0

So,

=> a = 6 or 6

=> a = 6

Thus by multiplying exponentials, we get, (a - 6)^3.

a^3 - 18a^2 + 108a - 216 = (a - 6)^3