Questions: For the polynomial function, (a) list all possible rational zeros, (b) find all rational zeros, and (c) factor f(x). f(x)=x^3+3x^2-25x+21 (a) Determine the possible rational zeros for f(x). Choose the correct answer below. A. 1,3,-7 B. ±1, ±3, ±7 C. ±1, ±3, ±7, ±21 D. 1,3,7,21 (b) Determine the rational zeros for f(x). Choose the correct answer below. A. -1,-3,7 B. ±1, ±3, ±7, ±21 C. 1 D. 1,3,-7 (c) The factored form of f(x)=x^3+3x^2-25x+21 is f(x)= . (Simplify your answer. Factor completely. Use integers or fractions for any numbers in the expression.)

Solution Steps

(a) To list all possible rational zeros, use the Rational Root Theorem, which states that any rational zero of the polynomial \( f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \) is a factor of the constant term \( a_0 \) divided by a factor of the leading coefficient \( a_n \). For the given polynomial \( f(x) = x^3 + 3x^2 - 25x + 21 \), the constant term is 21 and the leading coefficient is 1. Therefore, the possible rational zeros are the factors of 21.

(b) To find the actual rational zeros, evaluate the polynomial at each of the possible rational zeros found in part (a) and check which ones yield zero.

(c) Once the rational zeros are found, use polynomial division or synthetic division to factor the polynomial completely.

Using the Rational Root Theorem, we determine the possible rational zeros of the polynomial \( f(x) = x^3 + 3x^2 - 25x + 21 \). The constant term is 21 and the leading coefficient is 1. The factors of 21 are \( \pm 1, \pm 3, \pm 7, \pm 21 \). Therefore, the list of possible rational zeros is: \[ \text{Possible rational zeros: } \pm 1, \pm 3, \pm 7, \pm 21 \]

Next, we evaluate the polynomial at each of the possible rational zeros to find the actual rational zeros. The evaluations yield the following results:

• \( f(1) = 0 \)
• \( f(3) = 0 \)
• \( f(-7) = 0 \)

Thus, the rational zeros of the polynomial are: \[ \text{Rational zeros: } 1, 3, -7 \]

With the rational zeros identified, we can factor the polynomial \( f(x) \). The factored form based on the rational zeros is: \[ f(x) = (x - 1)(x - 3)(x + 7) \]

Final Answer