Final answer:
To determine the number of possible rational roots for the polynomial P(x)=10x³+6x²+4x+4, we use the Rational Root Theorem. After listing and reducing all possible factor combinations of the constant term and the leading coefficient, we find there are eight distinct possible rational roots.
Step-by-step explanation:
To find out how many possible rational roots there are for the polynomial P(x)=10x³+6x²+4x+4, we can use the Rational Root Theorem. This theorem states that any rational root, in its reduced form π/q, must have π as a factor of the constant term (in this case, 4), and q as a factor of the leading coefficient (here, 10).
Factors of 4 (constant term) are ±1, ±2, ±4. Factors of 10 (leading coefficient) are ±1, ±2, ±5, ±10. Therefore, we list all possible combinations of these factors.
- ±1/1, -1/1, 1/2, -1/2, 1/5, -1/5, 1/10, -1/10
- ±2/1, -2/1, 2/2, -2/2, 2/5, -2/5, 2/10, -2/10
- ±4/1, -4/1, 4/2, -4/2, 4/5, -4/5, 4/10, -4/10
After reducing these fractions, we have a total of eight distinct possible rational roots: ±1, ±2, ±4, ±1/5, ±2/5, and ±4/5.