Answer :
To find all zeros and their multiplicities for the function [tex]\( f(x) = x^5 + x^4 - 19x^3 + 17x^2 + 48x - 60 \)[/tex], we can follow a series of steps:
### Step 1: Apply the Rational Zero Theorem
The Rational Zero Theorem suggests that any rational zero of the polynomial is of the form [tex]\( \frac{p}{q} \)[/tex], where [tex]\( p \)[/tex] is a factor of the constant term (-60) and [tex]\( q \)[/tex] is a factor of the leading coefficient (1). This gives us the list of possible rational zeros:
[tex]\[ \pm 1, \pm 2, \pm 3, \pm 4, \pm 5, \pm 6, \pm 10, \pm 12, \pm 15, \pm 20, \pm 30, \pm 60 \][/tex]
### Step 2: Use Graphing Calculator to Estimate Zeros
Graph [tex]\( f(x) \)[/tex] on a calculator to visually estimate where the function crosses the x-axis. This will help identify likely candidates from our list. Assume, for example, that you notice potential zeros around some small integer values.
### Step 3: Verify Candidates Using Synthetic Division
Let's check some of these candidates using synthetic division to see if they are indeed zeros.
1. Try [tex]\( x = 2 \)[/tex]:
Perform synthetic division of [tex]\( f(x) \)[/tex] by [tex]\( x - 2 \)[/tex].
- Coefficients of [tex]\( f(x) \)[/tex]: 1, 1, -19, 17, 48, -60
- Perform the division process:
```
2 | 1 1 -19 17 48 -60
| 2 6 -26 -18 60
------------------------
1 3 -13 -9 30 0
```
- Remainder is 0, so [tex]\( x = 2 \)[/tex] is a zero. The quotient [tex]\( x^4 + 3x^3 - 13x^2 - 9x + 30 \)[/tex] is the remaining polynomial.
2. Try [tex]\( x = -3 \)[/tex]:
Repeat synthetic division with the new polynomial for [tex]\( x + 3 \)[/tex].
```
-3 | 1 3 -13 -9 30
| -3 0 -39 144
-------------------
1 0 -13 -48 90
```
- Remainder is not zero, so [tex]\( x = -3 \)[/tex] is not a zero.
3. Try [tex]\( x = 3 \)[/tex]:
```
3 | 1 3 -13 -9 30
| 3 18 15 18
---------------------
1 6 5 6 48
```
- Similar non-zero remainder, so [tex]\( x = 3 \)[/tex] is not a zero.
Continue testing remaining candidates systematically or graphically refine the candidates again using a more zoomed-in view.
### Step 4: Solve Remaining Polynomial
Once a zero is confirmed and the polynomial reduces, repeat the process with the remaining polynomial using either synthetic division or factoring to find other zeros, possibly solving a simpler quadratic using the quadratic formula if it factors down that far.
### Conclusion
By following these steps, we'll identify:
- Rational zeros identified as factors using synthetic division.
- Check quadratic solutions wherever required using formulas.
Keep testing points, refining from graph observations, and reduce until all roots are found. Remember to check for multiplicities by seeing how many times a division yields a zero remainder when testing with synthetic division!
This structured approach helps ensure you find all zeros and verify their multiplicities effectively.
### Step 1: Apply the Rational Zero Theorem
The Rational Zero Theorem suggests that any rational zero of the polynomial is of the form [tex]\( \frac{p}{q} \)[/tex], where [tex]\( p \)[/tex] is a factor of the constant term (-60) and [tex]\( q \)[/tex] is a factor of the leading coefficient (1). This gives us the list of possible rational zeros:
[tex]\[ \pm 1, \pm 2, \pm 3, \pm 4, \pm 5, \pm 6, \pm 10, \pm 12, \pm 15, \pm 20, \pm 30, \pm 60 \][/tex]
### Step 2: Use Graphing Calculator to Estimate Zeros
Graph [tex]\( f(x) \)[/tex] on a calculator to visually estimate where the function crosses the x-axis. This will help identify likely candidates from our list. Assume, for example, that you notice potential zeros around some small integer values.
### Step 3: Verify Candidates Using Synthetic Division
Let's check some of these candidates using synthetic division to see if they are indeed zeros.
1. Try [tex]\( x = 2 \)[/tex]:
Perform synthetic division of [tex]\( f(x) \)[/tex] by [tex]\( x - 2 \)[/tex].
- Coefficients of [tex]\( f(x) \)[/tex]: 1, 1, -19, 17, 48, -60
- Perform the division process:
```
2 | 1 1 -19 17 48 -60
| 2 6 -26 -18 60
------------------------
1 3 -13 -9 30 0
```
- Remainder is 0, so [tex]\( x = 2 \)[/tex] is a zero. The quotient [tex]\( x^4 + 3x^3 - 13x^2 - 9x + 30 \)[/tex] is the remaining polynomial.
2. Try [tex]\( x = -3 \)[/tex]:
Repeat synthetic division with the new polynomial for [tex]\( x + 3 \)[/tex].
```
-3 | 1 3 -13 -9 30
| -3 0 -39 144
-------------------
1 0 -13 -48 90
```
- Remainder is not zero, so [tex]\( x = -3 \)[/tex] is not a zero.
3. Try [tex]\( x = 3 \)[/tex]:
```
3 | 1 3 -13 -9 30
| 3 18 15 18
---------------------
1 6 5 6 48
```
- Similar non-zero remainder, so [tex]\( x = 3 \)[/tex] is not a zero.
Continue testing remaining candidates systematically or graphically refine the candidates again using a more zoomed-in view.
### Step 4: Solve Remaining Polynomial
Once a zero is confirmed and the polynomial reduces, repeat the process with the remaining polynomial using either synthetic division or factoring to find other zeros, possibly solving a simpler quadratic using the quadratic formula if it factors down that far.
### Conclusion
By following these steps, we'll identify:
- Rational zeros identified as factors using synthetic division.
- Check quadratic solutions wherever required using formulas.
Keep testing points, refining from graph observations, and reduce until all roots are found. Remember to check for multiplicities by seeing how many times a division yields a zero remainder when testing with synthetic division!
This structured approach helps ensure you find all zeros and verify their multiplicities effectively.
