Question

A fourth degree polynomial with five terms could have, at most, how many different linear factors?

Answer 1

Given that a linear factor is of the type (x - k) where the value k is called a "root" (or a zero) of the polynomial, then you can easily imagine that H(x - a)(a - b)(x - c)(x - d) has no choice but to end up as a polynomial that begins with a term in x^4. (H is just a scalar = normal number that becomes the "leading coefficient" of the polynomial, once you perform the multiplication that rebuilds the polynomial) The number of terms in the polynomial does not really matter. for example, something like (x+1)(x-1)(x+2)(x-2) = x^4 - 5x^2 + 4 It is still a "fourth degree" polynomial; It is still a "real" polynomial (the coefficients 1, -5 and 4 are all real numbers) If you imagine throwing in a 5th linear factor, as in (x - a)(x - b)(x - c)(x - d)(x - e) there is no way to avoid ending up with an x^5 It is no longer a "fourth degree" polynomial. Therefore, a fourth degree polynomial CANNOT have more than 4 linear factors. Does not matter how many terms it has.