Can you always use synthetic division for dividing polynomials? Explain.
No, you cannot always use synthetic division for dividing polynomials. Synthetic division can only be used specifically for dividing by linear factors of the form (x - a). Synthetic division is a simplified method that facilitates the division process when the divisor is a linear factor. However, if the divisor is not in the form (x - a), then synthetic division cannot be utilized.
For example, synthetic division can be used to divide the polynomial f(x) = 3x^3 - 5x^2 + 2x - 7 by the linear factor (x - 2). In this case, synthetic division greatly simplifies the process and allows for efficient division.
However, if the divisor is not in the form (x - a), such as when dividing by a quadratic factor or any other non-linear expression, synthetic division cannot be applied. In such cases, alternative division methods like long division or factoring may be required.