Question

If f, left bracket, x, right bracket, equals, x, cubed, minus, 6, x, squared, plus, 11, x, minus, 6f(x)=x 3 −6x 2 +11x−6 and x, minus, 1x−1 is a factor of f, left bracket, x, right bracketf(x), then find all of the zeros of f, left bracket, x, right bracketf(x) algebraically

Answer 1

If x - 1 is a factor of f(x), then using polynomial division, we determine the zeros of f(x) as 1, 2, and 3. We get the quadratic equation x^2 - 5x + 6, which factors to (x - 2)(x - 3), leading to the remaining zeros. The zeros of f(x) are 1, 2, and 3.

Step-by-step explanation:

If x - 1 is a factor of f(x) = x³ - 6x² + 11x - 6, then one of the zeros of f(x) is x = 1. To find the other zeros, we should perform polynomial division to divide f(x) by x - 1, simplifying the expression to a quadratic form.

Once we have the quadratic equation, we can find the remaining zeros using the quadratic formula.

To perform polynomial division, we would find that:

f(x) ÷ (x - 1) = x² - 5x + 6

The resulting quadratic can be factored further into:

(x - 2)(x - 3)

From here, it's easy to see that the remaining zeros of f(x) are x = 2 and x = 3.

Thus, the zeros of f(x) are 1, 2, and 3.