Question

If one zero of the polynomial p(x) = x³ – 6x² + 11x – 6 is 3, find the other two zeros.

Answer 1

1. Identify the linear factor

Since x = 3 is a given zero of the polynomial, according to the Factor Theorem, (x – 3) must be a linear factor of p(x).

2. Divide the polynomial

To find the remaining factors, divide p(x) by (x – 3) using polynomial long division or synthetic division:

`(x^3 - 6x^2 + 11x - 6)/(x - 3) = x^2 - 3x + 2`

This gives us the rewritten polynomial expression:

p(x) = (x – 3)(x² – 3x + 2)

3. Factorize the quadratic quotient 

Next, find the zeros of the remaining quadratic quotient by setting it equal to zero and splitting the middle term:

x² – 3x + 2 = 0

x² – 2x – x + 2 = 0

x(x – 2) – 1(x – 2) = 0

(x – 1)(x – 2) = 0

Solving for x gives:

x – 1 = 0

⇒ x = 1

x – 2 = 0

⇒ x = 2