Answer: To find the highest common factor (HCF) of two polynomials, we can use the Euclidean algorithm for polynomials. The Euclidean algorithm involves dividing the larger polynomial by the smaller polynomial and continuing the division process until the remainder is zero. The last non-zero remainder is the HCF of the two polynomials.
In this case, we have:
2x^3 + x^2 - 3x = (2x^3 - x^3) + x^2 - 3x
= x^3 + x^2 - 3x
Now, we divide x^3 + x^2 - 3x by x^3 - x
x^3 + x^2 - 3x = x^3 - x * (1 + x - 3)
= x^3 - x * (x^2 - 2x + 3)
We cannot simplify this expression further, so the HCF of 2x^3 + x^2 - 3x and x^3 - x is x^3 - x.
Explanation: