Question

Use factor theorem to verify that (3x + 1) is a factor of the polynomical


`3 {x}^(3 ) + 4 {x}^(2) - 5x - 2`
​

Answer 1

(3x + 1) is a factor of the polynomial

Explanation:

If (ax + b) is a factor of f(x) then f(-
`(b)/(a)` ) is a factor

Then if (3x + 1) is a factor then f (-
`(1)/(3)` ) = 0

substituting x = -
`(1)/(3)` into the polynomial

3 (-
`(1)/(3)` )³ + 4( -
`(1)/(3)` )² - 5 ( -
`(1)/(3)` ) - 2

= 3 (-
`(1)/(27)` ) + 4 (
`(1)/(9)` ) +
`(5)/(3)` - 2 ( change all values into ninths )

= -
`(1)/(9)` +
`(4)/(9)` +
`(15)/(9)` -
`(18)/(9)` ( evaluating in pairs )

=
`(3)/(9)` -
`(3)/(9)`

= 0

then (3x + 1) is a factor of the polynomial