# One root of f(x)= x^{3} - 9x^{2} + 26x - 24 is x = 2. What are all the roots of the function? Use the remainder theorem.

**Solution:**

f(x) = x^{3} - 9x^{2} + 26x - 24

One factor is x - 2

Using the remainder theorem, we know that if x-a is a factor of f(x), then f(a) is the remainder. Here f(2)= 0. When we divide f(x) by the factor, then the quotient is another factor of f(x).

It can be written as x = 2

If we divide f(x) by x - 2 using synthetic division, we get the quotient, which is the other factor.

Quotient = x^{2 }- 7x + 12 = 0

By splitting the middle terms

x^{2 }- 4x - 3x + 12 = 0

Taking out x as common in the first two terms and 3 as common in the last two terms

x (x - 4) - 3 (x - 4) = 0

(x - 4) (x - 3) = 0

x - 4 = 0 and x - 3 = 0

x = 4 and x = 3

Therefore, the roots of the function are x = 2, 4, and 3.

## One root of f(x)= x^{3} - 9x^{2} + 26x - 24 is x = 2. What are all the roots of the function? Use the remainder theorem.

**Summary:**

The roots of the function x^{3} - 9x^{2} + 26x - 24 are x = 2, 4 and 3.

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