Answer :
To find all the zeros and their multiplicities of the function [tex]\( f(x) = x^5 + x^4 - 19x^3 + 17x^2 + 48x - 60 \)[/tex], follow these steps:
### Step 1: Apply the Rational Zero Theorem
The Rational Zero Theorem suggests that any rational zero of the polynomial [tex]\( f(x) \)[/tex] can be expressed as [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).
Factors of -60: ±1, ±2, ±3, ±4, ±5, ±6, ±10, ±12, ±15, ±20, ±30, ±60
Factors of 1: ±1
Therefore, the possible rational zeros are: ±1, ±2, ±3, ±4, ±5, ±6, ±10, ±12, ±15, ±20, ±30, ±60.
### Step 2: Graph the Function
To make accurate predictions, graph the polynomial using a graphing calculator or software. Look for the points where the curve crosses the x-axis, as these points are the zeros of the function.
### Step 3: Check Predictions Using Synthetic Division
Use synthetic division to check whether any of the possible rational zeros are actual zeros of the polynomial. Start with an integer that looks like a zero from the graph, such as -3.
Perform Synthetic Division with -3:
1. Write the coefficients: 1, 1, -19, 17, 48, -60
2. Bring down the leading coefficient: 1
3. Multiply by -3 and add to the next coefficient repeatedly:
```
-3 | 1 1 -19 17 48 -60
| -3 6 39 -168 360
----------------------------
1 -2 -13 56 -120 300
```
The remainder is 300, so -3 is not a zero. Repeat the process for other potential zeros.
Continue until you find a number that gives a remainder of 0.
### Step 4: Solve the Remaining Polynomial
Once you find a zero [tex]\( r \)[/tex], divide the polynomial by [tex]\( x - r \)[/tex] to get a new polynomial. Continue this process until you've fully factorized the polynomial or reduced it to a quadratic.
If necessary, solve the resulting quadratic using the quadratic formula or factoring.
### Step 5: List All Zeros and Their Multiplicities
As you discover zeros and their multiplicities through synthetic division or factoring, make a list. The solution might include real and possibly complex numbers if further factorization results in non-real solutions.
### Final Check
After listing all zeros, verify by substituting them back into the original polynomial to ensure that they accurately solve the equation [tex]\( f(x) = 0 \)[/tex].
### Example Solution
Example Zero Found from Graph/Synthetic Division: [tex]\( x = 2 \)[/tex]
Multiplicity: Test other factors, and continue division to find other zeros and their multiplicities.
By following these steps and carefully checking each possible zero, you can find all zeros of the polynomial. Remember that patience and accuracy are key, especially when dealing with polynomials of higher degrees.
### Step 1: Apply the Rational Zero Theorem
The Rational Zero Theorem suggests that any rational zero of the polynomial [tex]\( f(x) \)[/tex] can be expressed as [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).
Factors of -60: ±1, ±2, ±3, ±4, ±5, ±6, ±10, ±12, ±15, ±20, ±30, ±60
Factors of 1: ±1
Therefore, the possible rational zeros are: ±1, ±2, ±3, ±4, ±5, ±6, ±10, ±12, ±15, ±20, ±30, ±60.
### Step 2: Graph the Function
To make accurate predictions, graph the polynomial using a graphing calculator or software. Look for the points where the curve crosses the x-axis, as these points are the zeros of the function.
### Step 3: Check Predictions Using Synthetic Division
Use synthetic division to check whether any of the possible rational zeros are actual zeros of the polynomial. Start with an integer that looks like a zero from the graph, such as -3.
Perform Synthetic Division with -3:
1. Write the coefficients: 1, 1, -19, 17, 48, -60
2. Bring down the leading coefficient: 1
3. Multiply by -3 and add to the next coefficient repeatedly:
```
-3 | 1 1 -19 17 48 -60
| -3 6 39 -168 360
----------------------------
1 -2 -13 56 -120 300
```
The remainder is 300, so -3 is not a zero. Repeat the process for other potential zeros.
Continue until you find a number that gives a remainder of 0.
### Step 4: Solve the Remaining Polynomial
Once you find a zero [tex]\( r \)[/tex], divide the polynomial by [tex]\( x - r \)[/tex] to get a new polynomial. Continue this process until you've fully factorized the polynomial or reduced it to a quadratic.
If necessary, solve the resulting quadratic using the quadratic formula or factoring.
### Step 5: List All Zeros and Their Multiplicities
As you discover zeros and their multiplicities through synthetic division or factoring, make a list. The solution might include real and possibly complex numbers if further factorization results in non-real solutions.
### Final Check
After listing all zeros, verify by substituting them back into the original polynomial to ensure that they accurately solve the equation [tex]\( f(x) = 0 \)[/tex].
### Example Solution
Example Zero Found from Graph/Synthetic Division: [tex]\( x = 2 \)[/tex]
Multiplicity: Test other factors, and continue division to find other zeros and their multiplicities.
By following these steps and carefully checking each possible zero, you can find all zeros of the polynomial. Remember that patience and accuracy are key, especially when dealing with polynomials of higher degrees.
