Question

Use the rational root theorem to list all possible roots for the equation. Then use synthetic division and factoring to find any actual rational roots.

10x^3+ 11x^2- 16x+4=0

Answer 1

The possible rational roots of the equation 10x^3 + 11x^2 - 16x + 4 = 0 are listed using the rational root theorem. Synthetic division or factoring can then be used to find any actual rational roots.

Step-by-step explanation:

To find the possible rational roots of the equation 10x^3 + 11x^2 - 16x + 4 = 0, we can use the rational root theorem. According to the theorem, the possible rational roots are of the form p/q, where p is a factor of the constant term (4) and q is a factor of the leading coefficient (10).

The factors of 4 are ±1, ±2, and ±4. The factors of 10 are ±1 and ±2. So, the possible rational roots are:

• ±1/1
• ±2/1
• ±4/1
• ±1/2
• ±2/2
• ±4/2

Now, we can use synthetic division or factoring to find any actual rational roots. By applying synthetic division with one of the possible rational roots, we can check if it results in a remainder of 0. If it does, then that root is an actual rational root of the equation. Otherwise, we move on to the next possible root.