Question

Math

Posted 3 months ago

```
Given the polynomials $p(z)=z^{\wedge} 4-16 z^{\wedge} 2+40 z-25$ and $d(z)=z^{\wedge} 2-4 z+5$.
a) Perform division of $p(z)$ by $d(z)$.
b) Determine from the result in a) all the roots of the polynomial $p(z)$.
```

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Answer from Sia

Posted 3 months ago

Solution by Steps

step 1

Given the polynomials $p(z) = z^4 - 16z^2 + 40z - 25$ and $d(z) = z^2 - 4z + 5$, we need to perform the division of $p(z)$ by $d(z)$

step 2

We start by expressing the division as $\frac{p(z)}{d(z)} = \frac{z^4 - 16z^2 + 40z - 25}{z^2 - 4z + 5}$

step 3

Using polynomial long division or synthetic division, we find that $\frac{z^4 - 16z^2 + 40z - 25}{z^2 - 4z + 5} = z^2 + 4z - 5$

step 4

Therefore, $p(z) = (z^2 - 4z + 5)(z^2 + 4z - 5)$

step 5

To find the roots of $p(z)$, we solve the equation $p(z) = 0$

step 6

This gives us $(z^2 - 4z + 5)(z^2 + 4z - 5) = 0$

step 7

Solving $z^2 - 4z + 5 = 0$ using the quadratic formula $z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, we get $z = 2 \pm i$

step 8

Solving $z^2 + 4z - 5 = 0$ using the quadratic formula, we get $z = -5$ and $z = 1$

Answer

The roots of the polynomial $p(z)$ are $z = -5$, $z = 1$, $z = 2 - i$, and $z = 2 + i$.

Key Concept

Polynomial Division and Root Finding

Explanation

We performed polynomial division to factorize $p(z)$ and then solved the resulting quadratic equations to find all the roots.

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