To find the exact roots of the equation x³ - 9x² + 26x - 2 = 0, we can use a numerical method called the Newton-Raphson method. This method allows us to approximate the roots iteratively.
1. Let's start by finding an initial guess for one of the roots. Looking at the equation, we can see that x = 1 is close to a root.
2. Using x = 1 as the initial guess, we can apply the Newton-Raphson method to refine the approximation. The formula for the next iteration is given by:
x(n+1) = x(n) - f(x(n))/f'(x(n))
3. We need to find the derivative of the function f(x) = x³ - 9x² + 26x - 2. The derivative is:
f'(x) = 3x² - 18x + 26
4. Now, we can plug in the values into the Newton-Raphson formula:
5. We have found an improved approximation for one of the roots: x ≈ -5/11.
To find the remaining roots, we can use polynomial long division or synthetic division to divide the given equation by the factor (x - (-5/11)). This will give us a quadratic equation that we can solve to find the remaining roots.
Performing polynomial long division or synthetic division, we get:
(x³ - 9x² + 26x - 2) ÷ (x + 5/11) = x² - 14x + 22
The quadratic equation x² - 14x + 22 = 0 can be solved using the quadratic formula or factoring.
Using the quadratic formula:
x = (-(-14) ± √((-14)² - 4(1)(22))) / (2(1))
x = (14 ± √(196 - 88)) / 2
x = (14 ± √108) / 2
x = (14 ± 2√27) / 2
x = 7 ± √27
Therefore, the exact roots of the equation x³ - 9x² + 26x - 2 = 0 are:
shadow or dark curtain describes a potentially urgent problem when vision is partially or totally blocked by dark or blurred shapes often beginning in the peripheral or side vision. This disturbance may come from above, below or from the side. It may occur in one or both eyes at the same time.
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Answer:
To find the exact roots of the equation x³ - 9x² + 26x - 2 = 0, we can use a numerical method called the Newton-Raphson method. This method allows us to approximate the roots iteratively.
1. Let's start by finding an initial guess for one of the roots. Looking at the equation, we can see that x = 1 is close to a root.
2. Using x = 1 as the initial guess, we can apply the Newton-Raphson method to refine the approximation. The formula for the next iteration is given by:
x(n+1) = x(n) - f(x(n))/f'(x(n))
3. We need to find the derivative of the function f(x) = x³ - 9x² + 26x - 2. The derivative is:
f'(x) = 3x² - 18x + 26
4. Now, we can plug in the values into the Newton-Raphson formula:
x(1) = 1 - (1³ - 9(1)² + 26(1) - 2)/(3(1)² - 18(1) + 26)
x(1) = 1 - (1 - 9 + 26 - 2)/(3 - 18 + 26)
x(1) = 1 - (16)/(11)
x(1) = -5/11
5. We have found an improved approximation for one of the roots: x ≈ -5/11.
To find the remaining roots, we can use polynomial long division or synthetic division to divide the given equation by the factor (x - (-5/11)). This will give us a quadratic equation that we can solve to find the remaining roots.
Performing polynomial long division or synthetic division, we get:
(x³ - 9x² + 26x - 2) ÷ (x + 5/11) = x² - 14x + 22
The quadratic equation x² - 14x + 22 = 0 can be solved using the quadratic formula or factoring.
Using the quadratic formula:
x = (-(-14) ± √((-14)² - 4(1)(22))) / (2(1))
x = (14 ± √(196 - 88)) / 2
x = (14 ± √108) / 2
x = (14 ± 2√27) / 2
x = 7 ± √27
Therefore, the exact roots of the equation x³ - 9x² + 26x - 2 = 0 are:
x ≈ -5/11, x ≈ 7 + √27, x ≈ 7 - √27
Answer:
shadow or dark curtain describes a potentially urgent problem when vision is partially or totally blocked by dark or blurred shapes often beginning in the peripheral or side vision. This disturbance may come from above, below or from the side. It may occur in one or both eyes at the same time.
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