Final answer:
To identify the integer roots of the polynomial equation 0.25x³ - 0.25x² - x + 1, first convert it to x³ - x² - 4x + 4. Then assess the possible roots using the Rational Root Theorem, considering factors of 4 as potential roots, and verify each potential root by substitution or synthetic division.
Step-by-step explanation:
To find all integer roots of the equation 0.25x³ - 0.25x² - x + 1, we need to examine the possible integer factors of the constant term, which in this case is 1. However, before we proceed to look for roots, it might be simpler to eliminate the decimals by multiplying through by a power of ten. Multiplying every term by 4 will convert our cubic equation into an integer coefficient polynomial: x³ - x² - 4x + 4.
By applying the Rational Root Theorem, we can assess which integers might be the roots of this polynomial. Given the coefficients, the possible integer roots are factors of 4, which are ±1, ±2, and ±4.
We can look for roots using substitution or synthetic division to test each possible root. If a number is a root, it will satisfy the polynomial equation making it equal to zero. Once we find a root, we can factor it out and reduce the cubic polynomial to a quadratic, for which we can apply the quadratic formula if needed.