Answer :
Certainly! Let's solve this step-by-step.
Given the polynomial equation:
[tex]\[ x^3 - 10x^2 - 13x + 22 = 0 \][/tex]
### Part (a) Find all potential rational solutions using the Rational Root Theorem
The Rational Root Theorem states that any potential rational solution of the polynomial equation \(a_n x^n + \cdots + a_1 x + a_0 = 0\) is of the form \( \frac{p}{q} \), where \( p \) is a factor of the constant term (\( a_0 \)) and \( q \) is a factor of the leading coefficient (\( a_n \)).
For the polynomial \( x^3 - 10x^2 - 13x + 22 \):
- The constant term \( a_0 \) is \( 22 \).
- The leading coefficient \( a_n \) is \( 1 \).
#### Step 1: Factors of the constant term (22)
The factors of \( 22 \) are:
[tex]\[ \pm 1, \pm 2, \pm 11, \pm 22 \][/tex]
#### Step 2: Factors of the leading coefficient (1)
The factors of \( 1 \) are:
[tex]\[ \pm 1 \][/tex]
#### Step 3: Form potential rational solutions \( \frac{p}{q} \)
Using the factors of the constant term and the factors of the leading coefficient, we get the potential rational solutions:
[tex]\[ \frac{1}{1}, \frac{2}{1}, \frac{11}{1}, \frac{22}{1}, \frac{-1}{1}, \frac{-2}{1}, \frac{-11}{1}, \frac{-22}{1} \][/tex]
Therefore, the potential rational solutions are:
[tex]\[ 1, 2, 11, 22, -1, -2, -11, -22 \][/tex]
### Part (b) Find all the distinct, actual rational solutions
To find the actual rational solutions, we need to test each potential solution by substituting it into the original polynomial equation \( x^3 - 10x^2 - 13x + 22 = 0 \) and see which ones satisfy the equation.
#### Test the potential solutions:
1. \( x = 1 \):
[tex]\[ 1^3 - 10(1)^2 - 13(1) + 22 = 1 - 10 - 13 + 22 = 0 \][/tex]
So, \( x = 1 \) is a root.
2. \( x = 2 \):
[tex]\[ 2^3 - 10(2)^2 - 13(2) + 22 = 8 - 40 - 26 + 22 = -36 \][/tex]
So, \( x = 2 \) is not a root.
3. \( x = 11 \):
[tex]\[ 11^3 - 10(11)^2 - 13(11) + 22 = 1331 - 1210 - 143 + 22 = 0 \][/tex]
So, \( x = 11 \) is a root.
4. \( x = 22 \):
[tex]\[ 22^3 - 10(22)^2 - 13(22) + 22 = 10648 - 4840 - 286 + 22 = 5544 \][/tex]
So, \( x = 22 \) is not a root.
5. \( x = -1 \):
[tex]\[ (-1)^3 - 10(-1)^2 - 13(-1) + 22 = -1 - 10 + 13 + 22 = 24 \][/tex]
So, \( x = -1 \) is not a root.
6. \( x = -2 \):
[tex]\[ (-2)^3 - 10(-2)^2 - 13(-2) + 22 = -8 - 40 + 26 + 22 = 0 \][/tex]
So, \( x = -2 \) is a root.
7. \( x = -11 \):
[tex]\[ (-11)^3 - 10(-11)^2 - 13(-11) + 22 = -1331 - 1210 + 143 + 22 = -2376 \][/tex]
So, \( x = -11 \) is not a root.
8. \( x = -22 \):
[tex]\[ (-22)^3 - 10(-22)^2 - 13(-22) + 22 = -10648 - 4840 + 286 + 22 = -19180 \][/tex]
So, \( x = -22 \) is not a root.
Thus, the actual rational solutions are:
[tex]\[ 1, 11, -2 \][/tex]
### Summary:
- (a) The potential rational solutions are: [tex]\[ 1, 2, 11, 22, -1, -2, -11, -22 \][/tex]
- (b) The actual rational solutions are: [tex]\[ 1, 11, -2 \][/tex]
Given the polynomial equation:
[tex]\[ x^3 - 10x^2 - 13x + 22 = 0 \][/tex]
### Part (a) Find all potential rational solutions using the Rational Root Theorem
The Rational Root Theorem states that any potential rational solution of the polynomial equation \(a_n x^n + \cdots + a_1 x + a_0 = 0\) is of the form \( \frac{p}{q} \), where \( p \) is a factor of the constant term (\( a_0 \)) and \( q \) is a factor of the leading coefficient (\( a_n \)).
For the polynomial \( x^3 - 10x^2 - 13x + 22 \):
- The constant term \( a_0 \) is \( 22 \).
- The leading coefficient \( a_n \) is \( 1 \).
#### Step 1: Factors of the constant term (22)
The factors of \( 22 \) are:
[tex]\[ \pm 1, \pm 2, \pm 11, \pm 22 \][/tex]
#### Step 2: Factors of the leading coefficient (1)
The factors of \( 1 \) are:
[tex]\[ \pm 1 \][/tex]
#### Step 3: Form potential rational solutions \( \frac{p}{q} \)
Using the factors of the constant term and the factors of the leading coefficient, we get the potential rational solutions:
[tex]\[ \frac{1}{1}, \frac{2}{1}, \frac{11}{1}, \frac{22}{1}, \frac{-1}{1}, \frac{-2}{1}, \frac{-11}{1}, \frac{-22}{1} \][/tex]
Therefore, the potential rational solutions are:
[tex]\[ 1, 2, 11, 22, -1, -2, -11, -22 \][/tex]
### Part (b) Find all the distinct, actual rational solutions
To find the actual rational solutions, we need to test each potential solution by substituting it into the original polynomial equation \( x^3 - 10x^2 - 13x + 22 = 0 \) and see which ones satisfy the equation.
#### Test the potential solutions:
1. \( x = 1 \):
[tex]\[ 1^3 - 10(1)^2 - 13(1) + 22 = 1 - 10 - 13 + 22 = 0 \][/tex]
So, \( x = 1 \) is a root.
2. \( x = 2 \):
[tex]\[ 2^3 - 10(2)^2 - 13(2) + 22 = 8 - 40 - 26 + 22 = -36 \][/tex]
So, \( x = 2 \) is not a root.
3. \( x = 11 \):
[tex]\[ 11^3 - 10(11)^2 - 13(11) + 22 = 1331 - 1210 - 143 + 22 = 0 \][/tex]
So, \( x = 11 \) is a root.
4. \( x = 22 \):
[tex]\[ 22^3 - 10(22)^2 - 13(22) + 22 = 10648 - 4840 - 286 + 22 = 5544 \][/tex]
So, \( x = 22 \) is not a root.
5. \( x = -1 \):
[tex]\[ (-1)^3 - 10(-1)^2 - 13(-1) + 22 = -1 - 10 + 13 + 22 = 24 \][/tex]
So, \( x = -1 \) is not a root.
6. \( x = -2 \):
[tex]\[ (-2)^3 - 10(-2)^2 - 13(-2) + 22 = -8 - 40 + 26 + 22 = 0 \][/tex]
So, \( x = -2 \) is a root.
7. \( x = -11 \):
[tex]\[ (-11)^3 - 10(-11)^2 - 13(-11) + 22 = -1331 - 1210 + 143 + 22 = -2376 \][/tex]
So, \( x = -11 \) is not a root.
8. \( x = -22 \):
[tex]\[ (-22)^3 - 10(-22)^2 - 13(-22) + 22 = -10648 - 4840 + 286 + 22 = -19180 \][/tex]
So, \( x = -22 \) is not a root.
Thus, the actual rational solutions are:
[tex]\[ 1, 11, -2 \][/tex]
### Summary:
- (a) The potential rational solutions are: [tex]\[ 1, 2, 11, 22, -1, -2, -11, -22 \][/tex]
- (b) The actual rational solutions are: [tex]\[ 1, 11, -2 \][/tex]
