Questions: Completely factor and find all zeros of the following polynomial. f(x)=-4x^4-19x^3+43x^2-95x+315 Write f as the product of linear factors, each with a leading coefficient of 1. f(x)=1 Find all zeros of f.

Solution Steps

To completely factor the polynomial and find all zeros, we will:

1. Use a numerical method to find the roots of the polynomial.
2. Once the roots are found, express the polynomial as a product of linear factors.

The polynomial given is

\[ f(x) = -4x^4 - 19x^3 + 43x^2 - 95x + 315. \]

Using numerical methods, we find the roots of the polynomial to be:

\[ r_1 = -7, \quad r_2 = 2.25, \quad r_3 = 8.6989 \times 10^{-16} + 2.2361i, \quad r_4 = 8.6989 \times 10^{-16} - 2.2361i. \]

The polynomial can be expressed as a product of linear factors based on its roots. The real roots contribute the following factors:

\[ f(x) = -4(x + 7)(x - 2.25)(x - (8.6989 \times 10^{-16} + 2.2361i))(x - (8.6989 \times 10^{-16} - 2.2361i)). \]

To express \(f(x)\) with a leading coefficient of 1, we can factor out \(-4\):

\[ f(x) = -4 \left( (x + 7)(x - 2.25) \left( (x - 8.6989 \times 10^{-16})^2 + (2.2361)^2 \right) \right). \]

Final Answer