Final answer:
To factor the polynomial given that k is a zero, we find factors of the constant term 35 and test combinations with the leading coefficient 2 to match the middle terms with coefficients 3 and -72. The possible linear factors include (2x - 5) and (x + 7), leading to a complete factorization of either (x - k)(2x - 5)(x + 7) or (x + k)(2x - 5)(x + 7) when including the given zero k.
Step-by-step explanation:
To factor f(x) into linear factors given that k is a zero of f(x), we must look for factors of the constant term that can combine with the leading coefficient to eventually give us all terms of the polynomial when multiplied out. Since the constant term is 35 and the leading coefficient is 2, we can test possible combinations of factors of 35 (±1, ±5, ±7, ±35) with 2 to see if we can obtain the middle terms with coefficients 3 and -72.
Testing the factors, we can match 2x and -5 to give us -10x and x and 7 to give us 7x, which sum up to -3x, the opposite of our middle term 3x. Likewise, the product of -5 and 7 gives us -35, which is again the opposite of our constant term 35. This suggests that possible linear factors include (2x - 5) and (x + 7).
Since the polynomial is of degree 3 and k is a zero, the remaining factor must be of the form (x - k) or (x + k). Thus, the complete factorization of the polynomial could be either (x - k)(2x - 5)(x + 7) or (x + k)(2x - 5)(x + 7), as those combinations yield the original polynomial when multiplied out.