Question

Determine the no. of zeros of the polynomial x4-1.​

Answer 1

x^4-1=0

X^4=1

x=1,-1,i,-i

there are two real roots and two unreal roots

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Answer 2

Real zeros of equation are 1 and -1 and complex zeros are i and -i

Solution:

Given polynomial equation is:

$x^4 - 1$

We have to find the number of zeros

From given, find zeros,

$x^4 - 1 = 0$

Rewrite as,

$(x^2)^2 - 1 = 0$

Use the following identity,

$a^2 - b^2 = (a+b)(a-b)$

Therefore,

$(x^2+1)(x^2 - 1) = 0$

Equate to 0,

$x^2 + 1 = 0\\\\x^2 = -1\\\\Take\ square\ root\ on\ both\ sides\\\\x = \pm i$

Also,

$x^2-1 = 0\\\\x^2 = 1\\\\Take\ square\ root\ on\ both\ sides\\\\x = \pm 1$

Thus, real zeros of equation are 1 and -1 and complex zeros are i and -i

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