One root of f(x)= x3 - 9x2 + 26x - 24 is x = 2. What are all the roots of the function? Use the remainder theorem.
Solution:
f(x) = x3 - 9x2 + 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 = x2 - 7x + 12 = 0
By splitting the middle terms
x2 - 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)= x3 - 9x2 + 26x - 24 is x = 2. What are all the roots of the function? Use the remainder theorem.
Summary:
The roots of the function x3 - 9x2 + 26x - 24 are x = 2, 4 and 3.