Questions: The function below has at least one rational zero. Use this fact to find all zeros of the function. f(x) = 2x^3 + 5x^2 + 5x + 3 If there is more than one zero, separate them with commas. Write exact values, not decimal approximations.

The function below has at least one rational zero. Use this fact to find all zeros of the function.

f(x) = 2x^3 + 5x^2 + 5x + 3

If there is more than one zero, separate them with commas. Write exact values, not decimal approximations.

Transcript text: The function below has at least one rational zero. Use this fact to find all zeros of the function. \[ f(x)=2 x^{3}+5 x^{2}+5 x+3 \] If there is more than one zero, separate them with commas. Write exact values, not decimal approximations.

Solution

Solution Steps

Step 1: List all possible rational zeros using the Rational Root Theorem

Possible rational zeros are the fractions p/q where p is a factor of 3 and q is a factor of 2. Thus, the possible rational zeros are: 1/2, 1, 3/2, 3, -1/2, -3, -1, -3/2.

Step 2: Test each possible rational zero

Testing 1/2: f(1/2) = 7 Testing 1: f(1) = 15 Testing 3/2: f(3/2) = 57/2 Testing 3: f(3) = 117 Testing -1/2: f(-1/2) = 3/2 Testing -3: f(-3) = -21 Testing -1: f(-1) = 1 Testing -3/2: f(-3/2) = 0

Step 3: Rational zeros found

The rational zeros are: -3/2.