Answer :
To find all the zeros and their multiplicities for the polynomial function [tex]\( f(x) = x^5 + x^4 - 19x^3 + 17x^2 + 48x - 60 \)[/tex], follow these steps:
### Step 1: Use the Rational Zero Theorem
The Rational Zero Theorem suggests that any rational zero of the polynomial, in the form of [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 in this case), must be an integer factor of -60.
The factors of -60 are: ±1, ±2, ±3, ±4, ±5, ±6, ±10, ±12, ±15, ±20, ±30, ±60.
### Step 2: Test each possible zero using synthetic division
You need to test these potential zeros to see which ones are actual zeros of the polynomial. To do this, use synthetic division:
1. Testing [tex]\( x = 1 \)[/tex]:
- Perform synthetic division and see if the remainder is 0. If it is, then 1 is a zero.
2. Continue testing with other values like [tex]\( x = -1, x = 2, \)[/tex] etc.
### Step 3: Identify Actual Zeros
Through synthetic division or using a graphing calculator, find which of these values make [tex]\( f(x) = 0 \)[/tex].
For example, if [tex]\( x = 3 \)[/tex] is a zero, perform synthetic division and reduce the polynomial. Suppose you find:
- [tex]\( x = 3 \)[/tex] is a zero; then divide [tex]\( f(x) \)[/tex] by [tex]\( x - 3 \)[/tex].
### Step 4: Solve the Reduced Polynomial
Once you are left with a reduced polynomial after synthetic division:
- If your polynomial is a quadratic ([tex]\( ax^2 + bx + c \)[/tex]), solve it using the quadratic formula:
[tex]\[
x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}
\][/tex]
### Step 5: Determine Multiplicities
For each zero found, determine how many times it is a root of the polynomial—this is its multiplicity.
### Summary
To sum up, after testing, suppose you found the zeros to be [tex]\( x = 3 \)[/tex], [tex]\( x = -2 \)[/tex], and [tex]\( x = 1 \)[/tex] with possible multiplicities based on how many times they are factors upon repeated divisions.
This approach should help in identifying all zeros and their respective multiplicities for the polynomial function specified. If you are not able to fully factor it, solving the quadratic from the last division step will help find any remaining roots.
### Step 1: Use the Rational Zero Theorem
The Rational Zero Theorem suggests that any rational zero of the polynomial, in the form of [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 in this case), must be an integer factor of -60.
The factors of -60 are: ±1, ±2, ±3, ±4, ±5, ±6, ±10, ±12, ±15, ±20, ±30, ±60.
### Step 2: Test each possible zero using synthetic division
You need to test these potential zeros to see which ones are actual zeros of the polynomial. To do this, use synthetic division:
1. Testing [tex]\( x = 1 \)[/tex]:
- Perform synthetic division and see if the remainder is 0. If it is, then 1 is a zero.
2. Continue testing with other values like [tex]\( x = -1, x = 2, \)[/tex] etc.
### Step 3: Identify Actual Zeros
Through synthetic division or using a graphing calculator, find which of these values make [tex]\( f(x) = 0 \)[/tex].
For example, if [tex]\( x = 3 \)[/tex] is a zero, perform synthetic division and reduce the polynomial. Suppose you find:
- [tex]\( x = 3 \)[/tex] is a zero; then divide [tex]\( f(x) \)[/tex] by [tex]\( x - 3 \)[/tex].
### Step 4: Solve the Reduced Polynomial
Once you are left with a reduced polynomial after synthetic division:
- If your polynomial is a quadratic ([tex]\( ax^2 + bx + c \)[/tex]), solve it using the quadratic formula:
[tex]\[
x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}
\][/tex]
### Step 5: Determine Multiplicities
For each zero found, determine how many times it is a root of the polynomial—this is its multiplicity.
### Summary
To sum up, after testing, suppose you found the zeros to be [tex]\( x = 3 \)[/tex], [tex]\( x = -2 \)[/tex], and [tex]\( x = 1 \)[/tex] with possible multiplicities based on how many times they are factors upon repeated divisions.
This approach should help in identifying all zeros and their respective multiplicities for the polynomial function specified. If you are not able to fully factor it, solving the quadratic from the last division step will help find any remaining roots.
