find all roots y=2x^4-2x^3-2x^2-2x-4

Question

trying to find the roots please help!

Answer 1

Given y = 2x^4-2x^3-2x^2-2x-4

By synthetic division,

y = 2x^4-2x^3-2x^2-2x-4 = (x+1)(x-2)(x^2+2)

Now x^2+2 = 0

Subtract 2 from each side.

x^2 = -2

Apply squre root on each side.

x = ±√2i

Roots of y = 2x^4-2x^3-2x^2-2x-4 is x = -1,2,±√2i.

Comments

Sorry.... typo mistake....

p/q	2	2	2	2
1	2	4	6	8
-1	2	0	2	0

 

Answer 2

Identify Rational Zeros :

Usually it is not practical to test all possible zeros of a polynomial function using only synthetic substitution. The Rational Zero Theorem can be used for finding the some possible zeros to test.

Rational Root Theorem, if a rational number in simplest form p/q is a root of the polynomial equation a_nx ⁿ + a_n – 1x ^{n – 1} + ... + a₁x + a₀ = 0, then p is a factor of a₀ and q is a factor if a_n.

The function is y = 2x ⁴ - 2x ³ - 2x ²- 2x - 4.

If p/q is a rational zero, then p is a factor of - 4 and q is a factor of 2.

The possible values of p are   ± 1,   ± 4 .

The possible values for q are ± 1, ± 2.

By the Rational Roots Theorem, the only possible rational roots are, p/q = ± 1, ± 1/2, ± 2, ± 4.

Make a table for the synthetic division and test possible real zeros.

p/q	2	- 2	- 2	- 2	- 4
1	2	0	- 2	- 4	8
2	2	2	2	2	0

Since, f(2) = 0, x = 2 is a zero. The depressed polynomial is  2x ³ + 2x ² +2x + 2 = 0.

Answer 3

If p/q is a rational zero, then p is a factor of 2 and q is a factor of 2.

By the Rational Roots Theorem, the only possible rational roots are, p/q = ± 1, ± 1/2, ± 2.

Make a table for the synthetic division and test possible real zeros.

p/q	2	2	2	2
1	2	4	6	8
3	2	8	26	80
-1	2	0	2	0

Since, f(-1) = 0, x = - 1 is a zero. The depressed polynomial is  2x ² + 2 = 0.

2(x ² + 1) = 0

 x ² + 1 = 0

x ² = - 1 = (± i )²

Extract square roots from each side.

⇒ x = ± i.

Therefore, tthe roots of the function are x = - 1, x = 2, x = i, and x = - i.

Comments

Sorry.... typo mistake....

p/q	2	2	2	2
1	2	4	6	8
-1	2	0	2	0