hw to solve equation in the real number system, x^4 +12x^3 +10x^2-9x+9x+22=0
x^4 +12x^3 +10x^2-9x+9x+22=0
I suspect a typo, why would you have two x terms without combining them?
To solve the equation x^4 + 12x^3 + 10x^2 - 9x + 9x + 22 = 0, you can follow these steps:
Step 1: Combine like terms: In the given equation, 9x and -9x are like terms. We can combine them to get x^4 + 12x^3 + 10x^2 + 22 = 0.
Step 2: Rearrange the equation: Write the equation in descending order of powers of x: x^4 + 12x^3 + 10x^2 + 22 = 0.
Step 3: Factor out common factors: There are no common factors that can be factored out from this equation.
Step 4: Use factoring techniques: Unfortunately, factoring this equation further is not possible as it does not have any common factors or follow any recognizable factoring patterns.
Step 5: Use the rational root theorem: The rational root theorem states that if an equation has a rational root (a solution that can be expressed as a fraction), then that root will be of the form p/q, where p is a factor of the constant term (22 in this case) and q is a factor of the leading coefficient (1 in this case).
Step 6: Test possible rational roots: To find the rational roots of the equation, you can try substituting the possible factors of 22 (p) divided by the factors of 1 (q). The factors of 22 are ±1, ±2, ±11, ±22, and the factors of 1 are ±1.
Try substituting these values into the equation and see if any of them make the equation equal to zero. If you find a value that makes the equation equal to zero, that value is a rational root.
Step 7: Use numerical methods: If you cannot find any rational roots using the rational root theorem or factoring techniques, you can use numerical methods such as graphing the equation or using iterative numerical techniques like the Newton-Raphson method to approximate the roots of the equation.
Note: The given equation x^4 + 12x^3 + 10x^2 + 22 = 0 may not have any real solutions as the degree of the equation is even and the coefficient of the highest power term is positive. However, it is always best to follow the steps mentioned above to exhaust all possible methods of solving the equation in the real number system.