Answer :
To solve the problem of finding all zeros and their multiplicities for the polynomial function [tex]\( f(x) = x^5 + x^4 - 19x^3 + 17x^2 + 48x - 60 \)[/tex], we can use a systematic approach. Let's go through the solution step by step.
### Step 1: Apply the Rational Zero Theorem
The Rational Zero Theorem suggests that possible rational zeros of a polynomial function are the factors of the constant term divided by the factors of the leading coefficient.
- Constant term: [tex]\(-60\)[/tex]
- Leading coefficient: [tex]\(1\)[/tex]
Thus, the possible rational zeros are the factors of [tex]\(-60\)[/tex], which include:
[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: Graph the Function
By graphing the function using a graphing calculator, we can visually inspect potential zeros.
### Step 3: Determine Zeros
Using the graph, we can identify some likely candidates for the zeros. In this case, from the analysis, we find:
1. The zero [tex]\( x = -5 \)[/tex]
2. The zero [tex]\( x = 2 \)[/tex]
### Step 4: Verify Zeros Using Synthetic Division
To confirm these values, we would perform synthetic division with each candidate zero identified:
- Check [tex]\( x = -5 \)[/tex]: When you divide the polynomial by [tex]\( x + 5 \)[/tex], the remainder is 0, thus confirming it as a zero.
- Check [tex]\( x = 2 \)[/tex]: When you divide the polynomial by [tex]\( x - 2 \)[/tex], the remainder is also 0, confirming this as a zero as well.
### Step 5: Solve Remaining Polynomials
After confirming the zeros, we find additional zeros from the polynomial through calculation or additional methods.
From a deeper calculation:
- Other found zeros are approximately [tex]\(-1.732\)[/tex] and [tex]\(1.732\)[/tex].
### Step 6: State All Zeros
Thus, the zeros of the polynomial [tex]\( f(x) \)[/tex] include:
- [tex]\( x = -5 \)[/tex] with a multiplicity of 1
- [tex]\( x = 2 \)[/tex] with a multiplicity of 1
- [tex]\( x \approx -1.732\)[/tex], a complex conjugate pair
- [tex]\( x \approx 1.732\)[/tex], a complex conjugate pair
### Conclusion
The function [tex]\( f(x) = x^5 + x^4 - 19x^3 + 17x^2 + 48x - 60 \)[/tex] has the zeros:
- [tex]\( -5 \)[/tex]
- [tex]\( 2 \)[/tex]
- Approximately [tex]\(-1.732\)[/tex]
- Approximately [tex]\(1.732\)[/tex]
Each zero reflects where the function crosses or touches the x-axis, describing the behavior of the function completely in terms of its roots.
### Step 1: Apply the Rational Zero Theorem
The Rational Zero Theorem suggests that possible rational zeros of a polynomial function are the factors of the constant term divided by the factors of the leading coefficient.
- Constant term: [tex]\(-60\)[/tex]
- Leading coefficient: [tex]\(1\)[/tex]
Thus, the possible rational zeros are the factors of [tex]\(-60\)[/tex], which include:
[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: Graph the Function
By graphing the function using a graphing calculator, we can visually inspect potential zeros.
### Step 3: Determine Zeros
Using the graph, we can identify some likely candidates for the zeros. In this case, from the analysis, we find:
1. The zero [tex]\( x = -5 \)[/tex]
2. The zero [tex]\( x = 2 \)[/tex]
### Step 4: Verify Zeros Using Synthetic Division
To confirm these values, we would perform synthetic division with each candidate zero identified:
- Check [tex]\( x = -5 \)[/tex]: When you divide the polynomial by [tex]\( x + 5 \)[/tex], the remainder is 0, thus confirming it as a zero.
- Check [tex]\( x = 2 \)[/tex]: When you divide the polynomial by [tex]\( x - 2 \)[/tex], the remainder is also 0, confirming this as a zero as well.
### Step 5: Solve Remaining Polynomials
After confirming the zeros, we find additional zeros from the polynomial through calculation or additional methods.
From a deeper calculation:
- Other found zeros are approximately [tex]\(-1.732\)[/tex] and [tex]\(1.732\)[/tex].
### Step 6: State All Zeros
Thus, the zeros of the polynomial [tex]\( f(x) \)[/tex] include:
- [tex]\( x = -5 \)[/tex] with a multiplicity of 1
- [tex]\( x = 2 \)[/tex] with a multiplicity of 1
- [tex]\( x \approx -1.732\)[/tex], a complex conjugate pair
- [tex]\( x \approx 1.732\)[/tex], a complex conjugate pair
### Conclusion
The function [tex]\( f(x) = x^5 + x^4 - 19x^3 + 17x^2 + 48x - 60 \)[/tex] has the zeros:
- [tex]\( -5 \)[/tex]
- [tex]\( 2 \)[/tex]
- Approximately [tex]\(-1.732\)[/tex]
- Approximately [tex]\(1.732\)[/tex]
Each zero reflects where the function crosses or touches the x-axis, describing the behavior of the function completely in terms of its roots.
