what is the product of the rational zeros of the following function?\
f(x) =x^3 -2x^2-11x+12
-12 to get 12 at the end but I will do the problem
try x = 1
1^3 - 2*1^2 - 11*1 +12 = 1-2-11+12 = 0
so x = 1 is a root
and
(x-1) is a factor so divide and get
(x-1)(x^2 - x - 12)
(x-1)(x-4)(x+3)
so roots at
x = 1 , 4 , -3
ahhaa i get it! can you help me with my other problems. because i know how to do them i just forget. but once i see them done i will remeber! thanks:)
I am running out of time, but you can do them
To find the product of the rational zeros of a polynomial function, we need to factorize the polynomial and look for the rational zeros.
Given the polynomial function f(x) = x^3 - 2x^2 - 11x + 12, we need to factorize it first.
One way to factorize this polynomial is to use synthetic division or polynomial long division to divide it by its linear factors. In this case, we can use synthetic division.
1. Begin by writing down the coefficients of the polynomial: 1, -2, -11, 12.
2. Since we are looking for rational zeros, the potential rational zeros can be derived from the factors of the constant term (12) divided by the factors of the leading coefficient (1). Therefore, the potential rational zeros are ±1, ±2, ±3, ±4, ±6, ±12.
3. Start with one of the potential rational zeros, let's say x = 1.
- Set up the synthetic division as follows: write the potential zero (1) on the left, and list the coefficients (1, -2, -11, 12) on the right.
- Perform the synthetic division. The last element in the bottom row will tell you if the potential zero is a zero or not. If it is, the polynomial can be written as a product of the divisor and quotient.
4. Repeat step 3 for each potential rational zero. If the synthetic division does not result in a remainder of zero, move on to the next potential zero.
Performing synthetic division with x = 1, we get:
1 | 1 -2 -11 12
| 1 -1 -12
|_________________
1 -1 -12 0
The remainder is 0, which means x = 1 is a zero of the polynomial.
The quotient after dividing f(x) by (x - 1) is x^2 - x - 12.
Now, we can factorize the quotient: x^2 - x - 12 = (x - 4)(x + 3).
Therefore, the factorized form of the polynomial f(x) = x^3 - 2x^2 - 11x + 12 is (x - 1)(x - 4)(x + 3).
From this factorization, we can see that the rational zeros of the polynomial are x = 1, x = 4, and x = -3.
To find the product of these rational zeros, simply multiply them together: 1 * 4 * (-3) = -12.
Therefore, the product of the rational zeros of f(x) = x^3 - 2x^2 - 11x + 12 is -12.