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Sagot :
Certainly! Let's solve the polynomial equation step-by-step.
Given polynomial equation:
[tex]\[x^3 - 2x^2 - x + 2 = 0\][/tex]
### Part a) Finding Potential Rational Solutions Using the Rational Root Theorem:
The Rational Root Theorem states that any potential rational root, \( \frac{p}{q} \), of the polynomial \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 = 0\) must be such that:
- \(p\) is a factor of the constant term \(a_0\).
- \(q\) is a factor of the leading coefficient \(a_n\).
In this polynomial:
- The constant term \(a_0 = 2\).
- The leading coefficient \(a_n = 1\).
#### Factors of the constant term (\(a_0 = 2\)):
The factors of 2 are \( \pm1, \pm2 \).
#### Factors of the leading coefficient (\(a_n = 1\)):
The factors of 1 are \( \pm1 \).
#### Potential Rational Roots:
The potential rational roots \( \frac{p}{q} \) are given by:
[tex]\[ \frac{\text{factors of } 2}{\text{factors of } 1} \][/tex]
Thus, the potential rational roots are:
[tex]\[ \pm1, \pm2 \][/tex]
So, the potential rational solutions are:
\( -2, -1, 1, 2 \).
### Part b) Finding the Distinct, Actual Rational Solutions:
We need to test each potential rational root in the original polynomial to determine which ones are actual solutions.
Substitute each potential rational root into the polynomial \(x^3 - 2x^2 - x + 2\) to check if they satisfy the equation:
1. For \(x = -2\):
[tex]\[ (-2)^3 - 2(-2)^2 - (-2) + 2 = -8 - 8 + 2 + 2 = -12 \neq 0 \][/tex]
So, \(x = -2\) is not a solution.
2. For \(x = -1\):
[tex]\[ (-1)^3 - 2(-1)^2 - (-1) + 2 = -1 - 2 + 1 + 2 = 0 \][/tex]
So, \(x = -1\) is a solution.
3. For \(x = 1\):
[tex]\[ 1^3 - 2(1)^2 - 1 + 2 = 1 - 2 - 1 + 2 = 0 \][/tex]
So, \(x = 1\) is a solution.
4. For \(x = 2\):
[tex]\[ 2^3 - 2(2)^2 - 2 + 2 = 8 - 8 - 2 + 2 = 0 \][/tex]
So, \(x = 2\) is a solution.
### Conclusion:
The actual rational solutions to the polynomial equation \(x^3 - 2x^2 - x + 2 = 0\) are:
b) [tex]\( -1, 1, 2 \)[/tex].
Given polynomial equation:
[tex]\[x^3 - 2x^2 - x + 2 = 0\][/tex]
### Part a) Finding Potential Rational Solutions Using the Rational Root Theorem:
The Rational Root Theorem states that any potential rational root, \( \frac{p}{q} \), of the polynomial \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 = 0\) must be such that:
- \(p\) is a factor of the constant term \(a_0\).
- \(q\) is a factor of the leading coefficient \(a_n\).
In this polynomial:
- The constant term \(a_0 = 2\).
- The leading coefficient \(a_n = 1\).
#### Factors of the constant term (\(a_0 = 2\)):
The factors of 2 are \( \pm1, \pm2 \).
#### Factors of the leading coefficient (\(a_n = 1\)):
The factors of 1 are \( \pm1 \).
#### Potential Rational Roots:
The potential rational roots \( \frac{p}{q} \) are given by:
[tex]\[ \frac{\text{factors of } 2}{\text{factors of } 1} \][/tex]
Thus, the potential rational roots are:
[tex]\[ \pm1, \pm2 \][/tex]
So, the potential rational solutions are:
\( -2, -1, 1, 2 \).
### Part b) Finding the Distinct, Actual Rational Solutions:
We need to test each potential rational root in the original polynomial to determine which ones are actual solutions.
Substitute each potential rational root into the polynomial \(x^3 - 2x^2 - x + 2\) to check if they satisfy the equation:
1. For \(x = -2\):
[tex]\[ (-2)^3 - 2(-2)^2 - (-2) + 2 = -8 - 8 + 2 + 2 = -12 \neq 0 \][/tex]
So, \(x = -2\) is not a solution.
2. For \(x = -1\):
[tex]\[ (-1)^3 - 2(-1)^2 - (-1) + 2 = -1 - 2 + 1 + 2 = 0 \][/tex]
So, \(x = -1\) is a solution.
3. For \(x = 1\):
[tex]\[ 1^3 - 2(1)^2 - 1 + 2 = 1 - 2 - 1 + 2 = 0 \][/tex]
So, \(x = 1\) is a solution.
4. For \(x = 2\):
[tex]\[ 2^3 - 2(2)^2 - 2 + 2 = 8 - 8 - 2 + 2 = 0 \][/tex]
So, \(x = 2\) is a solution.
### Conclusion:
The actual rational solutions to the polynomial equation \(x^3 - 2x^2 - x + 2 = 0\) are:
b) [tex]\( -1, 1, 2 \)[/tex].
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