Respuesta :
Answer:
D. -1,1,-2i, 2i
Step-by-step explanation:
We should have 4 roots since this is a 4th degree polynomial. We just have to watch since some of them may be multiple roots.
x^4+3x^2-4 = 0
Factoring this equation
(x^2 - 1) (x^2+4) = 0
We can factor the first term
(x-1) (x+1) (x^2+4) =0
Using the zero product property
x-1 =0 x+1 = 0 x^2+4 =0
x=1 x=-1 x^2 = -4
Taking the square root of each side
sqrt(x^2) =± sqrt(-4)
x =± sqrt(-1) sqrt(4)
x =±i *2
x = ±2i
Steps:
So firstly, I will be factoring by grouping. Firstly, what two terms have a product of -4x⁴ and a sum of 3x²? That would be -x² and 4x². Replace 3x² with -x² + 4x²:
[tex]0=x^4-x^2+4x^2-4[/tex]
Next, factor x⁴ - x² and 4x² - 4 separately. Make sure that they have the same quantity on the inside:
[tex]0=x^2(x^2-1)+4(x^2-1)[/tex]
Now you can rewrite it as:
[tex]0=(x^2+4)(x^2-1)[/tex]
However, we can simplify it even further. With the second factor, apply the difference of squares rule (x² - y² = (x + y)(x - y)):
[tex]x^2-1=(x+1)(x-1)\\\\0=(x^2+4)(x+1)(x-1)[/tex]
Now it's fully factored. With this, apply the zero product property and solve:
[tex]x^2+4=0\\x^2=-4\\x=\pm \sqrt{-4}=\pm \sqrt{4}i\\x=\pm\ 2i\\x=2i,-2i\\\\x+1=0\\x=-1\\\\x-1=0\\x=1[/tex]
Answer
In short, your answer is D. -1, 1, -2i, 2i