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Sagot :
To find the zeros of the function [tex]\( f(x) = x^3 - 4x^2 - x + 4 \)[/tex] and their multiplicities, we will follow these steps:
1. Find the roots of the polynomial by solving [tex]\( f(x) = 0 \)[/tex].
2. Determine the multiplicity of each root.
### Step 1: Find the roots
We need to solve the equation:
[tex]\[ x^3 - 4x^2 - x + 4 = 0 \][/tex]
To find the roots, we can use factorization or the Rational Root Theorem. The Rational Root Theorem states that any rational root of the polynomial [tex]\( a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \)[/tex] must be a factor of the constant term [tex]\( a_0 \)[/tex] divided by a factor of the leading coefficient [tex]\( a_n \)[/tex].
For the polynomial [tex]\( x^3 - 4x^2 - x + 4 \)[/tex]:
- The constant term [tex]\( a_0 \)[/tex] is 4.
- The leading coefficient [tex]\( a_n \)[/tex] is 1.
The possible rational roots are the factors of 4 divided by the factors of 1, which are:
[tex]\[ \pm 1, \pm 2, \pm 4 \][/tex]
We shall test these possible roots using synthetic division or direct substitution to see if they yield a zero.
Testing [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = 1^3 - 4(1)^2 - 1 + 4 = 1 - 4 - 1 + 4 = 0 \][/tex]
So, [tex]\( x = 1 \)[/tex] is a root.
### Step 2: Factorize using the root found
Since [tex]\( x = 1 \)[/tex] is a root, we can factor out [tex]\( (x - 1) \)[/tex] from [tex]\( x^3 - 4x^2 - x + 4 \)[/tex].
Using synthetic division to divide [tex]\( x^3 - 4x^2 - x + 4 \)[/tex] by [tex]\( x - 1 \)[/tex]:
```
1 | 1 -4 -1 4
| 1 -3 -4
-----------------
1 -3 -4 0
```
The quotient is [tex]\( x^2 - 3x - 4 \)[/tex]. Therefore:
[tex]\[ x^3 - 4x^2 - x + 4 = (x - 1)(x^2 - 3x - 4) \][/tex]
### Step 3: Factorize the quadratic expression
Next, we factor [tex]\( x^2 - 3x - 4 \)[/tex]:
[tex]\[ x^2 - 3x - 4 \][/tex]
By finding factors of [tex]\(-4\)[/tex] that add up to [tex]\(-3\)[/tex]:
[tex]\[ x^2 - 3x - 4 = (x - 4)(x + 1) \][/tex]
### Step 4: List all the roots and their multiplicities
Therefore, the polynomial can be fully factored as:
[tex]\[ x^3 - 4x^2 - x + 4 = (x - 1)(x - 4)(x + 1) \][/tex]
This means the roots of [tex]\( f(x) \)[/tex] are:
[tex]\[ x = 1, 4, -1 \][/tex]
Each of these roots has a multiplicity of 1 because they each appear once in the factorization of the polynomial.
### Answer:
The zeros of the function are [tex]\( x = 1, 4, -1 \)[/tex]. Each has a multiplicity of 1.
1. Find the roots of the polynomial by solving [tex]\( f(x) = 0 \)[/tex].
2. Determine the multiplicity of each root.
### Step 1: Find the roots
We need to solve the equation:
[tex]\[ x^3 - 4x^2 - x + 4 = 0 \][/tex]
To find the roots, we can use factorization or the Rational Root Theorem. The Rational Root Theorem states that any rational root of the polynomial [tex]\( a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \)[/tex] must be a factor of the constant term [tex]\( a_0 \)[/tex] divided by a factor of the leading coefficient [tex]\( a_n \)[/tex].
For the polynomial [tex]\( x^3 - 4x^2 - x + 4 \)[/tex]:
- The constant term [tex]\( a_0 \)[/tex] is 4.
- The leading coefficient [tex]\( a_n \)[/tex] is 1.
The possible rational roots are the factors of 4 divided by the factors of 1, which are:
[tex]\[ \pm 1, \pm 2, \pm 4 \][/tex]
We shall test these possible roots using synthetic division or direct substitution to see if they yield a zero.
Testing [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = 1^3 - 4(1)^2 - 1 + 4 = 1 - 4 - 1 + 4 = 0 \][/tex]
So, [tex]\( x = 1 \)[/tex] is a root.
### Step 2: Factorize using the root found
Since [tex]\( x = 1 \)[/tex] is a root, we can factor out [tex]\( (x - 1) \)[/tex] from [tex]\( x^3 - 4x^2 - x + 4 \)[/tex].
Using synthetic division to divide [tex]\( x^3 - 4x^2 - x + 4 \)[/tex] by [tex]\( x - 1 \)[/tex]:
```
1 | 1 -4 -1 4
| 1 -3 -4
-----------------
1 -3 -4 0
```
The quotient is [tex]\( x^2 - 3x - 4 \)[/tex]. Therefore:
[tex]\[ x^3 - 4x^2 - x + 4 = (x - 1)(x^2 - 3x - 4) \][/tex]
### Step 3: Factorize the quadratic expression
Next, we factor [tex]\( x^2 - 3x - 4 \)[/tex]:
[tex]\[ x^2 - 3x - 4 \][/tex]
By finding factors of [tex]\(-4\)[/tex] that add up to [tex]\(-3\)[/tex]:
[tex]\[ x^2 - 3x - 4 = (x - 4)(x + 1) \][/tex]
### Step 4: List all the roots and their multiplicities
Therefore, the polynomial can be fully factored as:
[tex]\[ x^3 - 4x^2 - x + 4 = (x - 1)(x - 4)(x + 1) \][/tex]
This means the roots of [tex]\( f(x) \)[/tex] are:
[tex]\[ x = 1, 4, -1 \][/tex]
Each of these roots has a multiplicity of 1 because they each appear once in the factorization of the polynomial.
### Answer:
The zeros of the function are [tex]\( x = 1, 4, -1 \)[/tex]. Each has a multiplicity of 1.
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