Find the roots of x3-x2-10x-8=0
The coefficients are:
1 -1 -10 -8
which do not add up to zero. So (x-1) is not a factor.
If we change the sign of the odd powers, we get:
-1 -1 10 -8
which adds up to zero. So (x+1) is a factor.
Divide x3-x2-10x-8 by (x+1) to get
x²-2x-8 and factorize to find the other zeroes.
To find the roots of the equation x^3 - x^2 - 10x - 8 = 0, we can use a numerical method called the Newton-Raphson method. This method helps us approximate the values of the roots iteratively. Here's how you can solve it step-by-step:
Step 1: Set up the equation
Rewrite the equation as f(x) = x^3 - x^2 - 10x - 8 = 0.
Step 2: Determine an initial guess
Select a value for x called the initial guess, denoted as x₀. This initial guess should be close to one of the roots. Let's use -2 as our initial guess in this example.
Step 3: Find the derivative of f(x)
Differentiate f(x) with respect to x to get f'(x), which represents the derivative of f(x). In this case, f'(x) = 3x^2 - 2x - 10.
Step 4: Apply the Newton-Raphson formula
Use the Newton-Raphson formula x₁ = x₀ - f(x₀) / f'(x₀) to find a new approximation for the root.
Iterate this formula until you reach a satisfactory accuracy or until you find all the roots.
Here's the step-by-step calculation:
Iteration 1:
x₀ = -2 (initial guess)
x₁ = x₀ - f(x₀) / f'(x₀)
= -2 - ( (-2)^3 - (-2)^2 - 10(-2) - 8 ) / ( 3(-2)^2 - 2(-2) - 10 )
= -2 - ( -8 + 4 + 20 - 8 ) / ( 12 + 4 - 10 )
= -2 - (8 + 4 + 20 - 8) / 6
= -2 - 24 / 6
= -2 - 4
= -6
Iteration 2:
x₀ = -6 (new approximation)
x₁ = x₀ - f(x₀) / f'(x₀)
= -6 - ( (-6)^3 - (-6)^2 - 10(-6) - 8 ) / ( 3(-6)^2 - 2(-6) - 10 )
= -6 - ( -216 + 36 + 60 - 8 ) / ( 3(36) + 12 - 10 )
= -6 - ( -216 + 36 + 60 - 8 ) / ( 108 + 12 - 10 )
= -6 - ( -128 ) / ( 110 )
= -6 + 1.1636
= -4.8364
Iteration 3:
x₀ = -4.8364 (new approximation)
x₁ = x₀ - f(x₀) / f'(x₀)
= -4.8364 - ( (-4.8364)^3 - (-4.8364)^2 - 10(-4.8364) - 8 ) / ( 3(-4.8364)^2 - 2(-4.8364) - 10 )
≈ -4.836410225
After subsequent iterations, you can keep using the formula until you reach the desired level of accuracy. Repeat the process with the new approximation until you achieve the desired precision.
Please note that finding the roots of polynomials can sometimes be challenging, especially if the equation does not have rational roots. In such cases, numerical methods like the Newton-Raphson method are useful.
To find the roots of the equation x^3 - x^2 - 10x - 8 = 0, we can use a method called the Rational Root Theorem and synthetic division.
1. First, we need to list all the possible rational roots of the equation. The possible rational roots are determined by the factors of the constant term (in this case, -8) divided by the factors of the leading coefficient (in this case, 1). The factors of -8 are ±1, ±2, ±4, and ±8, and the factors of 1 are ±1. So, the possible rational roots are ±1, ±2, ±4, and ±8.
2. We can start by trying the first possible root, which is 1. Now, we perform synthetic division using 1 as the test root:
1 | 1 -1 -10 -8
1 0 -10 -18
The remainder is -18, so 1 is not a root of the equation.
3. Next, we move on to the next possible root, which is -1. We perform synthetic division again using -1 as the test root:
-1 | 1 -1 -10 -8
-1 2 8 2
The remainder is 2, so -1 is not a root of the equation.
4. We continue this process for the rest of the possible rational roots. Doing this, we find that 2 is a root of the equation. The synthetic division using 2 as the test root is:
2 | 1 -1 -10 -8
2 2 -16 -56
The remainder is -56, so 2 is a root of the equation.
5. Now that we have found one root, we can use polynomial long division to divide the original equation by the linear factor corresponding to the root 2. Dividing x^3 - x^2 - 10x - 8 by (x - 2), we get:
(x - 2)(x^2 + x - 4)
6. To find the remaining roots, we set the quadratic factor (x^2 + x - 4) equal to zero and solve for x. We can either factor it, use the quadratic formula, or complete the square to find the remaining roots. In this case, using the quadratic formula:
x = (-1 ± sqrt(1^2 - 4(1)(-4))) / 2
= (-1 ± sqrt(1 + 16)) / 2
= (-1 ± sqrt(17)) / 2
Therefore, the remaining roots are (-1 ± sqrt(17)) / 2.
So, the roots of the equation x^3 - x^2 - 10x - 8 = 0 are 2 and (-1 ± sqrt(17)) / 2.