Questions: Completely factor and find all zeros of the following polynomial. f(x)=-2x^4-21x^3-54x^2-147x-280 Write f as the product of linear factors, each with a leading coefficient of 1. f(x)=

Completely factor and find all zeros of the following polynomial.

f(x)=-2x^4-21x^3-54x^2-147x-280

Write f as the product of linear factors, each with a leading coefficient of 1.

f(x)=

Transcript text: Completely factor and find all zeros of the following polynomial. \[ f(x)=-2 x^{4}-21 x^{3}-54 x^{2}-147 x-280 \] Write $f$ as the product of linear factors, each with a leading coefficient of 1. \[ f(x)= \] $\square$

Solution

Solution Steps

To factor the polynomial and find its zeros, we can use the following approach:

Use the Rational Root Theorem to identify possible rational roots.
Test these possible roots using synthetic division or direct substitution to find actual roots.
Once a root is found, use polynomial division to divide the polynomial by the corresponding linear factor.
Repeat the process on the quotient polynomial until it is completely factored into linear factors.
The zeros of the polynomial are the roots found during the factorization process.

Step 1: Identify the Roots

The roots of the polynomial $ f(x) = -2x^4 - 21x^3 - 54x^2 - 147x - 280 $ are found to be: \[ x = -8, \quad x = -\frac{5}{2}, \quad x = -\sqrt{7}i, \quad x = \sqrt{7}i \]

Step 2: Factor the Polynomial

The polynomial can be factored as follows: \[ f(x) = -(x + 8)(2x + 5)(x^2 + 7) \] Here, $ (x + 8) $ and $ (2x + 5) $ correspond to the real roots, while $ (x^2 + 7) $ accounts for the complex roots.