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P(x) = x³ - x² - 14 x + 24

factors of 24 are possible rational roots of P(x), if at all rational roots exist.

so roots are possibly : + 1, 2, 3, 4, 6,8, 12, 24 or -1, -2, -3, -4, -6, -8, -12, -24

check with 1, -1, 2 .. x = 2 => P(2) = 0

let, P(x) = (x - 2) [ x² - a x + (24/-2) ]

then compare coefficients of x² term or x term, then

-a x² - 2 x² = - 1 x² => a = -1

2 a x -12 x = - 14 x => a = -1

P(x) = ( x -2 ) ( x² +x-12) =

find factors of -12 that have difference of -1.

so -4 and 3 are roots of x² +x -12

P(x) = (x - 2 ) ( x + 4 ) ( x - 3)

factors of 24 are possible rational roots of P(x), if at all rational roots exist.

so roots are possibly : + 1, 2, 3, 4, 6,8, 12, 24 or -1, -2, -3, -4, -6, -8, -12, -24

check with 1, -1, 2 .. x = 2 => P(2) = 0

let, P(x) = (x - 2) [ x² - a x + (24/-2) ]

then compare coefficients of x² term or x term, then

-a x² - 2 x² = - 1 x² => a = -1

2 a x -12 x = - 14 x => a = -1

P(x) = ( x -2 ) ( x² +x-12) =

find factors of -12 that have difference of -1.

so -4 and 3 are roots of x² +x -12

P(x) = (x - 2 ) ( x + 4 ) ( x - 3)