Function zero problem

Question

find all rational zeros of this function

f(x) = 3x⁴ + 16x³ - 59x² - 400x - 400

Answer 1

Hello Michelle,

To find the rational zeros of the polynomial function f(x) = 3x⁴ + 16x³ - 59x² - 400x - 400 (equivalently, to solve the equation 3x⁴ + 16x³ - 59x² - 400x - 400 = 0), we begin by using the Rational Zeros Theorem:

Rational Zeros Theorem

Let f(x) be a polynomial function with integer coefficients. If the rational number p/q (where p and q have no common factors other than ±1) is a zero of f(x), then p is a factor of the constant term of f(x) and q is a factor of the lead coefficient of f(x).

As applied to your problem, this says that the possible rational zeros of

f(x) = 3x⁴ + 16x³ - 59x² - 400x - 400

will have the form p/q, where p is a factor of -400 and q is a factor of 3. First, list all the integer factors of the constant term -400.

p: ±1,±2,±4,±5,±8,±10,±16,±20,±25,±40,±50,±80,±100,±200,±400

The factors of the lead coefficient 3 are

q:±1,±3

The possible rational zeros of the polynomial are all numbers of the form p/q.

p/q: ±1,±2,±4,±5,±8,±10,±16,±20,±25,±40,±50,±80,±100,±200,±400,

±1/3, ±2/3, ±4/3, ±5/3, ±8/3, ±10/3, ±16/3, ±20/3, ±25/3, ±40/3, ±50/3, ±80/3, ±100/3,±200/3, ±400/3

To determine which (if any) of these possible rational zeros are actually zeros, we can use synthetic division. Recall that if you divide a polynomial f(x) by x - r and the remainder is 0, then f(r) = 0 (that is, r is a zero of f(x).) Let us check if 1 is a zero by synthetic division.

1 _| 3 16 -59 -400 -400 (Coefficients Row)

3 3 -2 -402 ("Work Row")

3 19 -2 -402 -802 <----Remainder ("Results Row")

The remainder is -802 (not zero), so 1 is not a zero of the polynomial. There are a total of 60 possible rational zeros in the list, and it would take a long time to check them all. I will skip ahead and try 5 next.

5 _| 3 16 -59 -400 -400

15 155 480 400

3 31 96 80 0

Since the remainder is zero, 5 is a zero of f(x). From the results of this division, we know that x - 5 is a factor of f(x), and the quotient (result of the division) is 3x³ + 31x² + 96x + 80. That is, f(x) may be written as

f(x) = (x - 5)(3x³ + 31x² + 96x + 80).

The remaining zeros of f(x) must be zeros of 3x³ + 31x² + 96x + 80, so we now carry out synthetic division on this polynomial. Synthetic division by -4 gives a remainder of 0, so -4 is also a zero. The quotient obtained is 3x² + 19x + 20, and the remaining zeros of f(x) must be zeros of this polynomial. That is, they must be solutions of the equation 3x² + 19x + 20 = 0. You can solve this using the quadratic formula. The solutions are complex numbers, so there are no more rational roots. Therefore, we conclude (finally) that the rational roots of f(x) are 5 and -4. Hope that helps! Let me know if you have any questions.

William