Final answer:
To solve the given polynomial equation, we need to find its roots. The steps involve factoring, checking for rational roots, testing the roots, and solving the resulting quadratic equation. In this case, the roots of the equation are x=2, x=-1, and x=5.
Step-by-step explanation:
To solve the polynomial equation in the form x³-4x²-7x+10=0, we need to find the roots of the equation. Here are the steps to solve it:
- Factor the equation if possible. In this case, we cannot factor it directly.
- Check for any rational roots using the Rational Root Theorem. The possible rational roots are the factors of the constant term (in this case, 10) divided by the factors of the leading coefficient (in this case, 1). For this equation, the possible rational roots are ±1, ±2, ±5, and ±10.
- Use synthetic division or long division method to test each possible root until the equation equals to zero. By testing all the possible rational roots, we find that x=2 is a root of the equation.
- Using the Factor theorem, divide the original equation by the corresponding factor (x-2) to obtain a quadratic equation.
- Solve the quadratic equation to find the remaining roots. By solving the quadratic equation, we find that x=-1 and x=5 are the remaining roots of the original equation.
Therefore, the roots of the equation x³-4x²-7x+10=0 are x=2, x=-1, and x=5.
Learn more about Solving polynomial equations