Question

What are the real and complex solutions of the polynomial equation?

Answer 1

\(27x^3+125=0\)

I got x=-5/3 so far, I just am confused on what to do next....

Answer 2

Did you factor the binomial?
Do you know which kind of factoring you need?

Answer 3

huh? Don't I need to divide using the synthetic division?

Answer 4

Hint: \(27x^3 + 125 = (3x)^3 + (5)^3\)

Answer 5

I got the x=-5/3 by doing this:
\(27x^3=-125\) -->subtract 125 both sides
\(x^3=-125/27\) -->divide both sides by 27
\(x=-5/3\) --> cube root both -125 and 27

.......is this correct?

Answer 6

This is not how you solve this kind of problem.
You need to factor the left side. Then set each factor equal to zero.
You will get three solutions.

Answer 7

The binomial you have is a sum of cubes. There is a special kind of factoring for a sum of cubes.
You may be familiar with a perfect square trinomial, and with the difference of two squares which are all special cases of factoring. This is another one of those special cases.

Answer 8

Here is how a sum of two cubes is factored:
\(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)

Answer 9

In your case, a = 3x, and b = 5.

Answer 10

okay

Answer 11

I am like really confused now......:(

Answer 12

Use the factoring formula I gave you ,and use 3x for a and 5 for b. You should get this:
\((3x)^3 + (5)^3 = (3x + 5)[(3x)^2 - (3x)(5) + (5)^2] \)

Answer 13

okay

Answer 14

Now you need to simplify what is inside the square brackets.

Answer 15

okay so 3x^2-15x+25?

Answer 16

\((3x+5)(3x^2-15x+25)\)

Answer 17

Almost.
Notice the first term is (3x)^2. The entire 3x quantity has to be squared. That means the 3 is also squared.

Answer 18

I mean in the second parentheses.

Answer 19

okay

Answer 20

\((3x+5)(\color{red}{9}x^2-15x+25) = 0\)

Answer 21

Got it?

Answer 22

not really........:( sorry I'm normally good at math but this is just confusing :'(

Answer 23

You are very close.
Look at the 9 in red above.

Answer 24

okay

Answer 25

In the square brackets, the first term was (3x)^2
That means the quantity 3x must be squared, so (3x)^2 = 3x * 3x = 9x^2

Answer 26

oh!

Answer 27

now I get it :D

Answer 28

Ok, now we have correctly factored it.
Now we set each factor equal to zero, and we solve each equation.

Answer 29

how do we do the \(9x^2-15x+25\)=0?

Answer 30

do we have to do that one by completing the square?

Answer 31

\((3x+5)(9x^2-15x+25)=0\)

\(3x+5 = 0 ~~~or~~~ 9x^2-15x+25=0\)

The left one is easy, and you know how to solve it.
The right one, you need th equadratic formula.

Answer 32

oh! I get it now lol :P thanks one sec

Answer 33

Yes, completing the square will also work, but I think the quadratic formula is easier.

Answer 34

\(x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Answer 35

okay.....

Answer 36

For the quadratic formula, a = 9, b = -15, c = 25

Answer 37

\(x = \dfrac{15 \pm \sqrt{-15^2 - 4(9)(25)}}{2(9)}\)

Answer 38

\(x = \dfrac{15 \pm \sqrt{ 225- 900}}{18}\)

Answer 39

Almost.

\(x = \dfrac{15 \pm \sqrt{(-15)^2 - 4(9)(25)}}{2(9)}\)

Answer 40

what's different?

Answer 41

You calculated it correctly, but you need the parentheses around the -15 when you square -15.

Answer 42

oh yeah.....haha whoops

Answer 43

\(-15^2 = -225\) This is the opposite of 15 squared.
\((-15)^2 = 225\) This is the number -15 squared.

Answer 44

yep....I get my mistake now lol whoopsies so now:
\(x = \dfrac{15 \pm \sqrt{-675}}{18}\)

Answer 45

how can I simplify that now?

Answer 46

\(x = \dfrac{15 \pm \sqrt{(-675)}}{18}\)
Yes, I got the same.

Answer 47

675 = 3 * 225, and 225 = 15^2

Answer 48

so would you just put \(3(15)^2\) in sqrt?

Answer 49

I don't know that part.....

Answer 50

Yes, but remember it is -675, so you need to take out i.

Answer 51

\[\sqrt{3(15)^2}*i\]

Answer 52

Yes, you are correct.

\(x = \dfrac{15 \pm i\sqrt{-675}}{18}\)

\(x = \dfrac{15 \pm i\sqrt{675}}{18}\)

\(x = \dfrac{15 \pm i\sqrt{3(15)^2}}{18}\)

\(x = \dfrac{15 \pm 15i\sqrt{3}}{18}\)

Answer 53

okay. is that the answer?

Answer 54

Sorry, I really have to go, so I'll just finish it. Please continue working and use my solution to check your work.

Answer 55

so \(x = \dfrac{15 \pm 15i\sqrt{3}}{18}\) and x=-5/3

Answer 56

or is it "or"

Answer 57

15 and 18 are both divisible by 3, so we can reduce the fraction.

Answer 58

okay thanks! is that all I have to do?

Answer 59

x = \(\dfrac{5 \pm 15i\sqrt{3}}{6}\)

Answer 60

\(x = \dfrac{15 \pm 15i\sqrt{3}}{18}\)

\(x = \dfrac{5 \pm 5i\sqrt{3}}{6}\)

Answer 61

Almost. The second 15 in the numerator also needs to be divided by 3.

Answer 62

oh! okay thank you so much for all of your help! I get it now........

Answer 63

:D

Answer 64

Then you can separate the plus and the minus and write as two solutions and also remember the simple solution you already had, the real one. All three solutions are separated by the word "or."

Answer 65

okay :) thanks again!

Answer 66

\(x = -\dfrac{5}{3} \) or \(x = \dfrac{5 + 5i\sqrt{3}}{6}\) or \(x = \dfrac{5 - 5i\sqrt{3}}{6}\)

Answer 67

You're welcome.

Answer 68

:D