Question

Can someone help solve this step by step

Answer 1

Solution.

Since the irrational roots occur in pairs, 1 + 5 is also a root. The factors

corresponding to these roots are

(x – 1 + 5 )( x – 1 − 5), i.e.(x – 1)

2

– 5

i.e. x2

– 2x – 4.

Dividing x4

– 5x3

+ 4x2

+ 8x – 8 by x2

– 2x – 4, we get the quotient x2

– 3x + 2.

Therefore x4

– 5x3

+ 4x2

+ 8x – 8 = (x

2

– 2x – 4)(x

2

– 3x + 2)

= (x

2

– 2x – 4)(x – 1)(x – 2)

The roots of the equation are 1 ± 5, 1, 2.

Answer 2

corresponding to these roots are

(x – 1 + 5 )( x – 1 − 5), i.e.(x – 1)

i.e. x2– 2x – 4.

Dividing x4– 5x+ 4x2+ 8x – 8 by x2– 2x – 4, we get the quotient x2– 3x + 2.

∴

x4– 5x3 + 4x2+ 8x – 8

= (2x– 2x – 4) (2– 3x + 2)

= (2– 2x – 4)(x – 1)(x – 2)

The roots of the equation are 1 ± 5, 1, 2.