Question

F(x) = X5 + 5x4 + 20x3 - 96x - 80 Has Some Of Its Zeros As X = -2, -1,2 What Would Be The Factorization Of F(x)? A. (x + 2)(x + 1)(x-2

Answer 1

Answer:

$\left(x+1\right)\left(x-2\right)\left(x+2\right)(x +2 - 4i)(x +2 + 4i)$

Step-by-step explanation:

Given

$f(x) = x^5 + 5x^4 + 20x^3 - 96x - 80$

$Zeros: -2, -1, 2$

Required

Factorization of f(x)

The given zeros imply that:

$x=-2$     $x =-1$    $x = 2$

This gives:

$x + 2 =0$        $x + 1 = 0$       $x - 2 = 0$

So, some factors are:

(x + 2), (x + 1) and (x - 2)

Divide f(x) by the factors, to get the other factors:

$\frac{x^5 + 5x^4 + 20x^3 - 96x - 80}{(x + 2)(x + 1) (x - 2)}$

Using a factorization calculator, we have:

$\frac{\left(x+1\right)\left(x-2\right)\left(x+2\right)\left(x^2+4x+20\right)}{(x + 2)(x + 1) (x - 2)}$

Cancel out common terms

$x^2+4x+20$

Using quadratic formula, we have:

$x = \frac{-b \± \sqrt{b^2 - 4ac}}{2a}$

Where

$a = 1; b =4; c = 20$

$x = \frac{-4 \± \sqrt{4^2 - 4*1*20}}{2*1}$

$x = \frac{-4 \± \sqrt{16 - 80}}{2*1}$

$x = \frac{-4 \± \sqrt{-64}}{2}$

Using complex notation

$\sqrt{-64}= 8i$

So:

$x = \frac{-4 \± 8i}{2}$

Simplify the fraction

$x = -2 \± 4i$

Split

$x = -2 + 4i \ or\ x = -2 - 4i$

Equate to 0

$x +2 - 4i = 0 \ or\ x +2 + 4i = 0$

The other factors are: (x +2 - 4i) and (x +2 + 4i)

Hence, the factorization of f(x) is:

$\left(x+1\right)\left(x-2\right)\left(x+2\right)(x +2 - 4i)(x +2 + 4i)$