Final answer:
To find the rational zeros of the polynomial and write it in factored form, you can use the Rational Root Theorem. The possible rational zeros are 1, -1, 2, -2, 3, -3, 5, -5, 6, -6, 10, -10, 15, -15, 30, and -30. The factored form of the polynomial is (x-1)(x-6)(x+5).
Step-by-step explanation:
To find the rational zeros of the polynomial, we can use the Rational Root Theorem. The Rational Root Theorem states that if a rational number P/Q is a zero of a polynomial, where P is a factor of the constant term and Q is a factor of the leading coefficient, then P/Q will be a zero of the polynomial. In this case, the constant term is 30 and the leading coefficient is 1. The factors of 30 are ±1, ±2, ±3, ±5, ±6, ±10, ±15, ±30. Therefore, the possible rational zeros of the polynomial are:
- x = 1/1
- x = -1/1
- x = 2/1
- x = -2/1
- x = 3/1
- x = -3/1
- x = 5/1
- x = -5/1
- x = 6/1
- x = -6/1
- x = 10/1
- x = -10/1
- x = 15/1
- x = -15/1
- x = 30/1
- x = -30/1
To write the polynomial in factored form, we can use these zeros to factor the polynomial. We can start by using synthetic division to divide the polynomial by (x - 1), which gives us a quotient of x^2-3x-30. Then we can factor x^2-3x-30 using the quadratic formula or by using factoring techniques. The factored form of the polynomial is (x-1)(x-6)(x+5).