The polynomial f(x) is given by f(x) = x^3 + 6x^2 + x - 4. Hence, find the exact roots of the equation f(x) = 0.
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ChatGPT
more than 4 weeks ago
To find the roots of the equation f(x) = 0, we need to solve the equation x^3 + 6x^2 + x - 4 = 0.
One way to find the roots is by factoring the polynomial. However, in this case, the polynomial is not easily factorable. Therefore, we can use numerical methods such as the Newton-Raphson method or synthetic division to approximate the roots.
Using the Newton-Raphson method, we start with an initial guess for the root and iterate using the formula:
x_(n+1) = x_n - f(x_n) / f'(x_n)
where f'(x) is the derivative of the function f(x).
By iterating this formula, we can find the roots of the equation f(x) = 0.
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Google Gemini AI
more than 4 weeks ago
Step 1: Factor the polynomial using Rational Root Theorem
- List the factors of the constant term (-4): ±1, ±2, ±4 - List the factors of the leading coefficient (1): ±1 - Try all possible combinations of the factors to find rational roots.
We find that x = 2 is a rational root.
Step 2: Perform Synthetic Division
``` 2 | 1 6 1 -4 \ 2 8 18 --- 1 8 9 14 ```
The last number in the bottom row (14) is the remainder, which is not zero. Therefore, x = 2 is not a root of f(x) = 0.
Step 3: Use the Rational Root Theorem Again
Since 2 is not a root, the numerator of the rational root must be a factor of 14. The factors of 14 are: ±1, ±2, ±7, ±14
Repeat the synthetic division process for each factor of 14. We find that x = -7 is a rational root.
Step 4: Factor the Reduced Polynomial
Using synthetic division, we divide f(x) by (x - (-7)):