Questions: Let g(x)=-2(3x+4)(x-1)(x-3)^2 be a polynomial function. a. Find all the roots/zeros/x-intercepts. b. Find the y-intercept of g.

Solution Steps

To solve the given polynomial function \( g(x) = -2(3x + 4)(x - 1)(x - 3)^2 \):

a. To find the roots/zeros/x-intercepts, set \( g(x) = 0 \) and solve for \( x \).

b. To find the y-intercept, evaluate \( g(0) \).

To find the roots of the polynomial function \( g(x) = -2(3x + 4)(x - 1)(x - 3)^2 \), we set \( g(x) = 0 \). The roots are obtained from the factors of the polynomial:

1. From \( 3x + 4 = 0 \), we find \( x = -\frac{4}{3} \).
2. From \( x - 1 = 0 \), we find \( x = 1 \).
3. From \( (x - 3)^2 = 0 \), we find \( x = 3 \) (with a multiplicity of 2).

Thus, the roots are: \[ x = -\frac{4}{3}, \quad x = 1, \quad x = 3 \]

To find the y-intercept of the function \( g \), we evaluate \( g(0) \): \[ g(0) = -2(3(0) + 4)(0 - 1)(0 - 3)^2 = -2(4)(-1)(9) = 72 \]

Final Answer