Answer :
To find all the zeros and their multiplicities for the function [tex]\( f(x) = x^5 + x^4 - 13x^3 - 17x^2 + 40x + 60 \)[/tex], we can follow these steps:
### Step 1: Apply the Rational Zero Theorem
The Rational Zero Theorem helps us find possible rational zeros. It states that any rational zero, expressed as [tex]\(\frac{p}{q}\)[/tex], will have [tex]\(p\)[/tex] (a factor of the constant term) and [tex]\(q\)[/tex] (a factor of the leading coefficient).
- The constant term of [tex]\(f(x)\)[/tex] is 60.
- The leading coefficient is 1.
Factors of 60 (±): [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]
Factors of 1 (±): [tex]\( \pm 1 \)[/tex]
Thus, the possible rational zeros are:
[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
Using a graphing calculator, plot the function to see where it crosses the x-axis. This helps estimate some potential zeros. Based on the graph, you might guess that some zeros are around certain integer points, such as [tex]\(x = 3\)[/tex].
### Step 3: Verify with Synthetic Division
Using synthetic division, confirm the zeros guessed from the graph.
#### Check if [tex]\(x = 3\)[/tex] is a zero:
Perform synthetic division of the polynomial [tex]\(f(x)\)[/tex] by [tex]\(x - 3\)[/tex]:
1. Coefficients: [tex]\(1, 1, -13, -17, 40, 60\)[/tex]
2. Place 3 (the guessed zero) as the divisor.
3. Use synthetic division to evaluate.
The synthetic division will give a remainder of 0 if [tex]\(x = 3\)[/tex] is indeed a zero:
- Results from synthetic division show: [tex]\([1, 4, -1, -20, -20, 0]\)[/tex]
The remainder is 0, confirming [tex]\(x = 3\)[/tex] is a zero.
### Step 4: Factor the Reduced Polynomial
The synthetic division also reveals the reduced polynomial:
[tex]\[ x^4 + 4x^3 - x^2 - 20x - 20 \][/tex]
You can continue to find further zeros of this reduced polynomial by checking other potential rational zeros or using numerical methods or other factoring techniques.
### Conclusion:
- Confirmed Zero: [tex]\( x = 3 \)[/tex]
- Other Rational Zeros to Check: Continue checking the reduced polynomial for additional rational or irrational zeros using similar methods. Each identified zero should be verified for its multiplicity (i.e., if a zero is repeated) using further division or graph inspection.
Using these methods, continue until all zeros and their multiplicities are found.
### Step 1: Apply the Rational Zero Theorem
The Rational Zero Theorem helps us find possible rational zeros. It states that any rational zero, expressed as [tex]\(\frac{p}{q}\)[/tex], will have [tex]\(p\)[/tex] (a factor of the constant term) and [tex]\(q\)[/tex] (a factor of the leading coefficient).
- The constant term of [tex]\(f(x)\)[/tex] is 60.
- The leading coefficient is 1.
Factors of 60 (±): [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]
Factors of 1 (±): [tex]\( \pm 1 \)[/tex]
Thus, the possible rational zeros are:
[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
Using a graphing calculator, plot the function to see where it crosses the x-axis. This helps estimate some potential zeros. Based on the graph, you might guess that some zeros are around certain integer points, such as [tex]\(x = 3\)[/tex].
### Step 3: Verify with Synthetic Division
Using synthetic division, confirm the zeros guessed from the graph.
#### Check if [tex]\(x = 3\)[/tex] is a zero:
Perform synthetic division of the polynomial [tex]\(f(x)\)[/tex] by [tex]\(x - 3\)[/tex]:
1. Coefficients: [tex]\(1, 1, -13, -17, 40, 60\)[/tex]
2. Place 3 (the guessed zero) as the divisor.
3. Use synthetic division to evaluate.
The synthetic division will give a remainder of 0 if [tex]\(x = 3\)[/tex] is indeed a zero:
- Results from synthetic division show: [tex]\([1, 4, -1, -20, -20, 0]\)[/tex]
The remainder is 0, confirming [tex]\(x = 3\)[/tex] is a zero.
### Step 4: Factor the Reduced Polynomial
The synthetic division also reveals the reduced polynomial:
[tex]\[ x^4 + 4x^3 - x^2 - 20x - 20 \][/tex]
You can continue to find further zeros of this reduced polynomial by checking other potential rational zeros or using numerical methods or other factoring techniques.
### Conclusion:
- Confirmed Zero: [tex]\( x = 3 \)[/tex]
- Other Rational Zeros to Check: Continue checking the reduced polynomial for additional rational or irrational zeros using similar methods. Each identified zero should be verified for its multiplicity (i.e., if a zero is repeated) using further division or graph inspection.
Using these methods, continue until all zeros and their multiplicities are found.
