Questions: Solve the polynomial equation. x^4 + x^3 = 10 - 5x - 3x^2 The solution(s) is/are (Simplify your answer. Type an exact answer, using radicals as needed. Express complex numbers in terms of i. Use a comma to separate answers as needed.)

Solve the polynomial equation.
x^4 + x^3 = 10 - 5x - 3x^2

The solution(s) is/are
(Simplify your answer. Type an exact answer, using radicals as needed. Express complex numbers in terms of i. Use a comma to separate answers as needed.)

Transcript text: Solve the polynomial equation. \[ x^{4}+x^{3}=10-5 x-3 x^{2} \] The solution(s) is/are $\square$ (Simplify your answer. Type an exact answer, using radicals as needed. Express complex numbers in terms of $i$. Use a comma to separate answers as needed.)

Solution

Solution Steps

To solve the polynomial equation, first rearrange all terms to one side of the equation to set it to zero. Then, simplify and combine like terms to form a standard polynomial equation. Use a numerical method or a polynomial solver to find the roots of the equation, which may include real and complex solutions.

Step 1: Rearranging the Equation

The given polynomial equation is

\[ x^{4} + x^{3} = 10 - 5x - 3x^{2}. \]

Rearranging all terms to one side, we have

\[ x^{4} + x^{3} + 3x^{2} + 5x - 10 = 0. \]

Step 2: Finding the Roots

To find the roots of the polynomial equation, we solve

\[ x^{4} + x^{3} + 3x^{2} + 5x - 10 = 0. \]

The solutions to this equation are

\[ x = -2, \quad x = 1, \quad x = -\sqrt{5}i, \quad x = \sqrt{5}i. \]