What is the completely factored form of f(x) = x3 – 2x2 – 5x + 6?
Asked by maham237 @ in Mathematics viewed by 322 People
Answered by maham237 @
Answer:
f(x) = (x -1)(x + 2)(x -3) is the factored form.
Step-by-step explanation:
The given function is f(x) = x³ -2x² - 5x + 6
We have to write it in the completely factored form.
For this we will use the rational roots theorem first.
In the equation x³ - 2x² - 5x + 6 = 0
P = ± multiples of constant term 6 = ± 1, ±2, ±3, ±6
Q = ± multiples of the coefficient of highest degree term = ±1
By theorem factors will be P/Q
Possible rational roots will be 1, ±2, ±3, ±6
Therefore 1 is a confirm rational root. Now we will find the depressed polynomial from synthetic division to find the other rational roots.
1 | 1 -2 -5 6
1 -1 -6
-------------------------
1 -1 -6 0
So the depressed polynomial will be (x² - x - 6).
Now can easily factorize this polynomial to get the rational roots.
x² - x - 6 = x² - 3x + 2x - 6
= x(x - 3) + 2(x - 3) = (x+2)(x-3)
Therefore whole factorized form of the polynomial function will be
f(x) = (x - 1)(x + 2)(x - 3)
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