Find the zeros of the function
x^5 - 3x^4 - 15x^3 + 45x^2 - 16x + 48 = 0
3,4,i
4,-4,i,-i
3,4,-4,i,-i
3,-4,-i
To find the zeros of the function, we can either use synthetic division or a graphing calculator.
Using synthetic division:
We will start by trying the potential zero x = 3.
3 | 1 -3 -15 45 -16 48
| 3 0 -45 0 -48
_______________________
1 0 -15 0 -16 0
The remainder is 0, so x - 3 is a factor of the function. Hence, (x - 3) is a zero.
Next, we divide the resulting polynomial by x - 3 using synthetic division:
(x - 3)(x^4 - 15x^2 - 16) = 0
Now, we have a quadratic equation:
x^4 - 15x^2 - 16 = 0
To solve the quadratic equation, we can let u = x^2:
u^2 - 15u - 16 = 0
Factoring the quadratic equation:
(u - 16)(u + 1) = 0
Setting each factor to zero:
u - 16 = 0 or u + 1 = 0
Solving for u in each case:
u = 16 or u = -1
Since u = x^2, we can solve for x:
x^2 = 16 or x^2 = -1
Taking the square root of both sides:
x = ±4 or x = ±i
Therefore, the zeros of the function are x = 3, 4, ±i.
The correct option is:
3, 4, ±i