find all the zeros of each equation x^5-3x^4-15x^3+45x^2-16x+48=0
x^5-3x^4-15x^3+45x^2-16x+48 = (x-3)(3x+1)(x-4)(x+4)
To find all the zeros of the equation x^5 - 3x^4 - 15x^3 + 45x^2 - 16x + 48 = 0, you can use various methods such as factoring, synthetic division, or using a numerical method like Newton's method. Let's use the Rational Root Theorem and synthetic division to find the zeros.
Step 1: Apply the Rational Root Theorem.
The Rational Root Theorem states that if a polynomial equation has a rational root (x = p/q), then p must be a factor of the constant term (48) and q must be a factor of the leading coefficient (1).
The factors of 48 are ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±16, ±24, and ±48.
The factors of 1 are ±1.
Step 2: Perform synthetic division.
Since this is a fifth-degree polynomial, we will find a rational root and divide the polynomial by (x - r), where r is a rational root.
Let's start with the first factor, ±1.
Synthetic division for x = 1:
1 | 1 - 3 - 15 + 45 - 16 + 48
| 1 - 2 - 13 + 32 + 16
____________________________
1 - 2 - 13 + 32 + 64
The result is 1x^4 - 2x^3 - 13x^2 + 32x + 64.
Synthetic division for x = -1:
-1 | 1 - 3 - 15 + 45 - 16 + 48
| -1 + 4 + 11 - 34 + 50
___________________________
1 - 4 - 11 + 11 + 98
The result is 1x^4 - 4x^3 - 11x^2 + 11x + 98.
Now, we need to find the rational root(s) for the remaining quartic equation we obtained above.
Step 3: Repeat the process for the remaining quartic equation.
Now, you can apply the Rational Root Theorem again to find the rational roots of the remaining quartic equations: 1x^4 - 2x^3 - 13x^2 + 32x + 64 and 1x^4 - 4x^3 - 11x^2 + 11x + 98.
Continue dividing until there are no rational roots left or until you obtain a quadratic equation that can be solved using the quadratic formula.
Note: While synthetic division helps us find rational roots efficiently, it doesn't guarantee that we find all the zeros of the equation. In some cases, we may need to resort to numerical methods to find all the zeros.
To find the zeros of the equation x^5 - 3x^4 - 15x^3 + 45x^2 - 16x + 48 = 0, we can use the Rational Root Theorem paired with synthetic division.
Step 1: Apply the Rational Root Theorem
The Rational Root Theorem states that if a polynomial equation has a rational root (i.e., a zero) in the form p/q, where p is a factor of the constant term (48) and q is a factor of the leading coefficient (1), then p/q is a possible zero of the equation.
The factors of 48 are ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±16, ±24, ±48.
The factors of 1 (the leading coefficient) are ±1.
Therefore, the possible rational zeros are:
p/q = ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±16, ±24, ±48.
Step 2: Use Synthetic Division to Test the Possible Zeros
We will now use synthetic division to test each possible zero to determine which ones are the actual zeros of the equation.
Let's start with the first possible zero, p/q = -1:
Perform synthetic division with -1 as the test zero:
```
-1 | 1 -3 -15 45 -16 48
-1 4 11 -56 72
----------------------------
1 -4 -11 -11 56 120
```
The remainder is 120, which is not zero. Therefore, -1 is not a zero of the equation.
Repeat the same process for the remaining possible zeros:
```
2 | 1 -3 -15 45 -16 48
2 -2 -26 38 44
-------------------------
1 -1 -17 19 22 92
3 | 1 -3 -15 45 -16 48
3 0 -45 90 90
-------------------------
1 0 -15 45 74 138
4 | 1 -3 -15 45 -16 48
4 4 36 124 32
---------------------
1 1 21 169 16 80
6 | 1 -3 -15 45 -16 48
6 18 18 198 168
------------------------
1 3 3 243 152 216
8 | 1 -3 -15 45 -16 48
8 40 200 440 352
-----------------------
1 5 25 485 336 400
12 | 1 -3 -15 45 -16 48
12 72 114 558 504
------------------------
1 9 57 603 488 552
16 | 1 -3 -15 45 -16 48
16 208 376 548 484
-------------------------
1 13 193 533 532
24 | 1 -3 -15 45 -16 48
24 504 998 1108 2672
-------------------------
1 21 489 1153 2720
```
From the synthetic division, we can see that the remainders are not zero for all the possible zeros tested. Therefore, none of the possible zeros (±1, ±2, ±3, ±4, ±6, ±8, ±12, ±16, ±24, ±48) are actual zeros of the equation.
Hence, this equation does not have any rational zeros.
To find the other zeros, you can use numerical methods such as graphing or using techniques like the Newton-Raphson method to approximate the zeros.