Questions: The function below has at least one rational zero. Use this fact to find all zeros of the function. g(x)=7x^3+19x^2+3x-5 If there is more than one zero, separate them with commas.

Solution Steps

To find the zeros of the polynomial function \( g(x) = 7x^3 + 19x^2 + 3x - 5 \), we can use the Rational Root Theorem to identify possible rational zeros. The Rational Root Theorem suggests that any rational zero, expressed as a fraction \( \frac{p}{q} \), has \( p \) as a factor of the constant term and \( q \) as a factor of the leading coefficient. We will test these possible rational zeros using synthetic division or direct substitution to find actual zeros. Once a zero is found, we can factor the polynomial and solve for the remaining zeros.

We are given the polynomial function \( g(x) = 7x^3 + 19x^2 + 3x - 5 \). Our goal is to find all the zeros of this polynomial.

Using numerical methods, we find the zeros of the polynomial. The calculated zeros are approximately:

• \( -2.4142 \)
• \( -0.7143 \)
• \( 0.4142 \)

The zeros can be expressed as:

• \( z_1 \approx -2.4142 \)
• \( z_2 \approx -0.7143 \)
• \( z_3 \approx 0.4142 \)

Final Answer