To factor the polynomial 35x^3 - 40x^2 + 21x - 24 completely, we can use various methods such as factoring by grouping or using the rational root theorem.
In this case, we can use the rational root theorem to find the possible rational roots of the polynomial. The rational root theorem states that if a polynomial has a rational root p/q (where p is a factor of the constant term and q is a factor of the leading coefficient), then p is a factor of the constant term and q is a factor of the leading coefficient.
For the given polynomial 35x^3 - 40x^2 + 21x - 24, the constant term is -24 and the leading coefficient is 35. The factors of -24 are ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±24, and the factors of 35 are ±1, ±5, ±7, ±35.
By testing the possible rational roots (using synthetic division or long division), we find that the polynomial does not have any rational roots.
Therefore, the polynomial 35x^3 - 40x^2 + 21x - 24 cannot be factored further using rational roots. It is already factored completely