We appreciate your visit to Find all of the zeros and their multiplicities of the function tex f x x 5 x 4 13x 3 17x 2 40x 60 tex. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!
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.
Thanks for taking the time to read Find all of the zeros and their multiplicities of the function tex f x x 5 x 4 13x 3 17x 2 40x 60 tex. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!
- Why do Businesses Exist Why does Starbucks Exist What Service does Starbucks Provide Really what is their product.
- The pattern of numbers below is an arithmetic sequence tex 14 24 34 44 54 ldots tex Which statement describes the recursive function used to..
- Morgan felt the need to streamline Edison Electric What changes did Morgan make.
Rewritten by : Barada
