Question

Medal and all that Junk. What are real and complex solutions of this polynomial equation? 27x^3+125=0

Answer 1

sorry idk :(

Answer 2

http://www.algebra.com/algebra/homework/Polynomials-and-rational-expressions.faq.question.393464.html

Answer 3

\(27x^3+125 = (3x)^3 + (5)^3\), right?

That is sum of cubes \(a^3 + b^3 = (a + b)(a^2 – ab + b^2) \)

Answer 4

Refresh pafe if you see question marks.

Answer 5

page*

Answer 6

Okay, yes. I see that.

Answer 7

OK, factoring given polynomial equation, you have \((3x + 5)(9x^2 + 15x + 25)=0\)

So either \(3x+5=0\) or \(9x^2 + 15x + 25=0\)

So you now have two equations to solve x for.

For second equation, you can apply quadratic formula.

Answer 8

oops, \(9x^2-15x+25\)***

Answer 9

I see that!! Alright, I'll solve out, can you check?

Answer 10

Sure.

Answer 11

So the first is x=-5/3 right?

Answer 12

Yes.

Answer 13

Okay. Give me a second for the second.

Answer 14

I got it down to 15+- sqrt of -675/16

Answer 15

/18*****

Answer 16

Now I know I'll need to use an imaginary form of this, but how would i go about that?

Answer 17

Well, \(\sqrt{-675}=i\sqrt{675}\), right?

Answer 18

Yep.

Answer 19

hey i gave u the answer

Answer 20

I know. @amysparkly12 but I wanted to figure out how to do each step!!

Answer 21

And I am sure you need to simplify \(\sqrt{675}\)

You can see that it is divisible by 25, so \(675= 25\times27 = 5^2\cdot3^3\)

So \(i\sqrt{675} = 5\cdot3i\sqrt{3} = 15i\sqrt{3}\). Make sense so far?

Answer 22

Sort of, why don't I just find the sqrt of 675 at that point and divide by 18?

Answer 23

So you have \(\dfrac{15\pm15i\sqrt3}{18}\)

You can factor 3 out and have them cancel: \(\dfrac{3(5\pm5i\sqrt3)}{3\cdot6}\)

You can do that, but if whatever inside square root is not square number, you will get messy number. See: \(\sqrt{675}=25.98076211353316...\)

Answer 24

And we don't want messy number in our hands, right?

Answer 25

Ah, I see now. Okay so after you factor out 3??

Answer 26

Obviously, we then cancel 3 out.

So we are left with \(x =\dfrac{5\pm5i\sqrt3}{6}\)

So you have two complex numbers as answers: \(x=\dfrac{5+5i\sqrt3}{6}\) and \(x=\dfrac{5-5i\sqrt3}{6}\)

Make sense so far?

Answer 27

Yes.

Answer 28

Now we just solve these two out?

Answer 29

Or are these our answers?

Answer 30

Well, you can arrange numbers around, but I think they are in their simplest form.

You are done here.

Answer 31

Awesommmmeeeee. Gah. Okay, thank you so much!

Answer 32

No problem!