Using factor theorem, factorize each of the following polynomials : `x^4-2x^3-7x^2+8x+12`

Explanation

`f(x) = x^4 – 2x^3 – 7x^2 + 8x + 12`
Constant term = 12 Factors of 12 are,` ±1, ±2, ±3, ±4, ±6, ±12`
Let `x – 1 = 0 or x = 1, f(1) = (1)^4 – 2(1)^3 – 7(1)^2 + 8(1) + 12`
`=> 1 – 2 – 7 + 8 + 12 = 12 ,f(1) =12`
Let `x + 1 = 0 or x = -1 , f(-1) = (-1)^4 – 2(-1)^3 – 7(-1)^2 + 8(-1) + 12`
`=> 1 + 2 – 7 – 8 + 12 = 0 , f(-1) = 0`
Let `x + 2 = 0 or x = -2 ,f(-2) = (-2)^4 – 2(-2)^3 – 7(-2)^2 + 8(-2) + 12`
`=> 16 + 16 – 28 – 16 + 12 = 0`
`f(-2) = 0 , x – 2 = 0 or x = 2, f(2) = (2)^4 – 2(2)^3 – 7(2)^2 + 8(2) + 12`
` => 16 – 16 – 28 + 16 + 12 = 0`
` f(2) = 0, x – 3 = 0 or x = 3 ,f(3) = (3)^4 – 2(3)^3 – 7(3)^2 + 8(3) + 12 = 0`
`f(3) = 0 `
Therefore, `(x – 1), (x + 2), (x – 2) \and\ (x - 3)` are factors of `f(x)`
Hence `f(x) = (x – 1)(x + 2) (x – 2) (x - 3)`