Question

Factor the expression 5x^3 - x^2 + 5x - 1

Answer 1

`5 x^(3) - x^(2) + 5x - 1`

Group like terms

`( 5x^(2) - x^(2) ) + (5x - 1)`

Find the common factor of each term and simplify

`x^(2) (5x - 1) + 1 (5x - 1)`

`( x^(2) + 1) (5x - 1)`




Additional :-
Find the roots of each of the terms:
(i)
`( x^(2) + 1)`
- using the quadratic equation (
`x = \frac{-b (+ or -) \sqrt{ b^(2) - 4ac } }{2a}`)
-
`\frac{-0 + \sqrt{0^(2 - 4(1)(1)) } }{2}` OR
`\frac{-0 - \sqrt{0^(2 - 4(1)(1)) } }{2}`
- Since the discriminant is negative (
`√(-4)`) [a negative number cannot be rooted] then this equation has no real roots (imaginary roots)

(ii)
`5x - 1`
- Simply Solve for x
5x - 1 = 0
5x = 1
x =
`(1)/(5)`

Thus the only solution this has or the only root this expression has is
`(1)/(5)` OR 0.2

Answer 2

Hello,

5x^3-x^3+5x-1=x²(5x-1)+(5x-1)=(5x-1)(x²+1)=(5x-1)(x-i)(x+i)