Question

What are all of the real roots of the following polynomial? f(x) x^4-24x^2-25

Answer 1

The real roots are ±5

Explanation:

It is given that,

f(x) = x^4 - 24x^2 - 25

To find the real roots

Let x^4 - 24x^2 - 25 = 0 ----(1)

Take y = x^2

Then eq (1) becomes,

y^2 - 24y - 25 = 0

By using splitting method we can write,

y^2 + y - 25y - 25 = 0

y(y + 1) - 25(y +1) = 0

(y + 1)(y - 25) = 0

(x^2 + 1 )(x^2 - 25) = 0

From (x^2 + 1 ) we get complex roots

x^2 = -1

x = √-1

x - 25 = 0 we get real roots

x = 25

x = ±5

Therefore the real roots are ±5

Answer 2

x = ±5

Explanation:

We are given the following polynomial function and we are to find all of its real roots:

`x^4-24x^2-25`

Let
`y=x^2` so we can now write it as:

`y^2-24y-25`

Factorizing it to get:

`(y^2+y)+(-25y-25)`

`y\left(y+1\right)-25\left(y+1\right)`

`\left ( y + 1 \right ) \left ( y - 2 5 \right)`

Substitute back
`y=x^2` to get:

`\left ( x^2 + 1 \right ) \left ( x^2 - 25 \right )`

`\left ( x^2 + 1 \right ) \left ( x + 5 \right ) \left ( x - 5 \right )`

The quadratic factor has only complex roots. Therefore, the real roots are x = ±5.