Math  /  Algebra

Question4. [0.31/0.43 Points] DETAILS MY NOTES SCOLALG7 3.5.034.MI. 3/100 Submissions Used PREVIOUS ANSWERS ASK YOUR TEACHE
Factor the polynomial completely. P(x)=x5+2x3P(x)=\begin{array}{r} P(x)=x^{5}+2 x^{3} \\ P(x)=\square \end{array} x= with multiplicity x= with multiplicity x= with multiplicity \begin{array}{l} x=\square \text { with multiplicity } \square \\ x=\square \text { with multiplicity } \square \\ x=\square \text { with multiplicity } \square \end{array} Need Help? Read II Whitch it Master It

Studdy Solution

STEP 1

What is this asking? We're asked to completely factor a polynomial and identify its roots along with their multiplicities. Watch out! Don't forget to consider complex roots and their multiplicities!

STEP 2

1. Factor out common terms.
2. Factor the remaining quadratic.

STEP 3

Alright, let's **factor** this polynomial!
We've got P(x)=x5+2x3P(x) = x^5 + 2x^3.
Notice that both terms have xx multiplied in there, and the smallest power of xx is x3x^3.
This means we can **factor out** an x3x^3 !

STEP 4

Let's do it!
Factoring out x3x^3 gives us P(x)=x3(x2+2)P(x) = x^3(x^2 + 2).
Boom! We've got a nice x3x^3 out front and a quadratic expression left inside the parentheses.

STEP 5

Now, let's tackle that quadratic, x2+2x^2 + 2.
To **find the roots**, we set it equal to zero: x2+2=0x^2 + 2 = 0.

STEP 6

Subtracting 2 from both sides gives us x2=2x^2 = -2.
Now, to **isolate** xx, we take the square root of both sides.
Remember, when we take the square root of a negative number, we get an imaginary number involving ii, where i=1i = \sqrt{-1}.

STEP 7

So, we have x=±2x = \pm\sqrt{-2}.
This simplifies to x=±2ix = \pm\sqrt{2} \cdot i.
This gives us two **complex roots**: x=2ix = \sqrt{2}i and x=2ix = -\sqrt{2}i.
Each of these roots has a multiplicity of **one**.

STEP 8

Don't forget about the x3x^3 we factored out earlier!
This gives us another root: x=0x = 0.
And since it's xx cubed, this root has a multiplicity of **three**.

STEP 9

P(x)=x3(x2+2)=x3(x2i)(x+2i)P(x) = x^3(x^2 + 2) = x^3(x-\sqrt{2}i)(x+\sqrt{2}i). x=0x = 0 with multiplicity 3 x=2ix = \sqrt{2}i with multiplicity 1 x=2ix = -\sqrt{2}i with multiplicity 1

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