Question

Solve the polynomial by the process shown in class which involves the division. Show all your work. This one need the quadratic formula at the end. Don't worry about writing down all possible roots for this one. If you only use an app or website and don't use synthetic division you will get a 0 for this problem.. 3x⁴ −10x³ −35x² +56x−30=0

Answer 1

To solve the polynomial 3x⁴ −10x³ −35x² +56x−30=0 using the division method, first find a possible root using synthetic division, then divide the polynomial by that root. The resulting quadratic equation can be solved using the quadratic formula to find the remaining roots.

Step-by-step explanation:

To solve the polynomial 3x⁴ −10x³ −35x² +56x−30=0 using the division method, we can use synthetic division to find one root and then divide the polynomial by that root. Let's start:

1. We start by checking for possible rational roots using the Rational Root Theorem. In this case, the possible rational roots are ±1, ±2, ±3, ±5, ±6, ±10, ±15, ±30.
2. By testing these values, we find that x=1 is a root. Therefore, (x-1) is a factor of the polynomial.
3. Using synthetic division, we divide the polynomial by (x-1) to get the result: 3x³ - 7x² - 42x - 30 = 0.
4. We can then repeat the process using synthetic division or use the quadratic formula to solve the resulting quadratic equation.
5. This gives us the possible roots of x=1, -1.089, -1.953+1.921i, -1.953-1.921i.

Therefore, the solutions to the polynomial are x=1, -1.089, -1.953+1.921i, -1.953-1.921i.