use remainder therem to find the remainder when f(x) is divided by x+2 then use factor to determine whether x +2 is a factor
f(x) = x^6 + 2x^5 + 3x^2 - 2x - 16
f(-2) = (-2)^6 + 2(-2)^5 + 3(-2)^2 -2(-2) - 16
= 64 -64 + 12 + 4 -16
= 0
so what do you think?
It's actually what I got and it is a factor because it's zero right?
To find the remainder when f(x) is divided by (x+2) using the Remainder Theorem, you can use polynomial long division or synthetic division. I will explain how to use synthetic division.
Step 1: Set up the synthetic division table by writing down the coefficients of the terms of f(x) in descending order.
-2 | 1 2 3 -2 -16
____________________
1
Step 2: Bring down the first coefficient (1).
-2 | 1 2 3 -2 -16
____________________
1
Step 3: Multiply the divisor, -2, by the result obtained in the previous step and write it under the next coefficient. Add these two values to get the new result.
-2 | 1 2 3 -2 -16
_________________
1 -2
______
1
Step 4: Repeat step 3 until you have gone through all the coefficients.
-2 | 1 2 3 -2 -16
_________________
1 -2 -2
______
1 0
Step 5: The final result, 1, is the remainder when f(x) is divided by (x+2).
To determine if x+2 is a factor of f(x), we check if the remainder is zero. If the remainder is zero, then the divisor is a factor. Since the remainder is 1, x+2 is not a factor of f(x).