Find all the zeros of each equation.
x^5-3x^4-15x^3+45x^2-16x+48=0
To find the zeros of the equation, we can use the Rational Root Theorem and synthetic division.
The Rational Root Theorem states that if a polynomial equation with integer coefficients has a rational root, then that root can be written as a fraction, where the numerator is a divisor of the constant term and the denominator is a divisor of the leading coefficient.
In this case, the constant term is 48 and the leading coefficient is 1. The possible rational roots are the divisors of 48, which are ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±16, ±24, and ±48.
Let's test these values using synthetic division:
For x = 1:
1 | 1 -3 -15 45 -16 48
| 1 -2 -17 28 12
___________________
1 -2 -17 28 60
For x = -1:
-1 | 1 -3 -15 45 -16 48
| -1 4 11 -34 50
___________________
1 1 -4 11 2
For x = 2:
2 | 1 -3 -15 45 -16 48
| 2 -2 -26 38 44
___________________
1 -1 -17 83 92
For x = -2:
-2 | 1 -3 -15 45 -16 48
| -2 10 10 -70 152
____________________
1 -1 -5 -25 200
For x = 3:
3 | 1 -3 -15 45 -16 48
| 3 0 -45 90 42
___________________
1 0 -15 135 90
For x = -3:
-3 | 1 -3 -15 45 -16 48
| -3 18 -9 -108 312
____________________
1 15 -24 -63 360
We have exhausted all possible rational roots. None of the values tested resulted in a remainder of 0, so there are no rational roots of this equation.
We can now use a graphing calculator or computer software to find the approximate zeros of the equation. The zeros are approximately x = -0.092, 0.905, 2.696, and 3.624.
Therefore, the zeros of the equation x^5 - 3x^4 - 15x^3 + 45x^2 - 16x + 48 = 0 are approximately x = -0.092, 0.905, 2.696, and 3.624.