Verify Whether the following are roots of the polynomial equations indicated against them.

x³ – 2x² – 5x + 6 = 0; x = 1, −2, 3

x³ – 2x² – 5x + 6 = 0

let p(x) = x³ – 2x² – 5x + 6

p(1) = (1)³ – 2(1)² – 5(1) + 6

= 1 – 2 × 1 – 5 + 6

= 1 – 2 – 5 + 6

= 7 – 7 = 0

∴ x = 1 is a root of x³ – 2x² – 5x + 6 = 0.

p(x) = x³ – 2x² – 5x + 6

p(-2) = (-2)³ – 2(-2)² – 5(-2) + 6

= -8 – 2 × 4 – 5 × 2 + 6

= -8 – 8 + 10 + 6

= -16 + 16 = 0

∴ x = -2 is a root of x³ – 2x² – 5x + 6 = 0.

p(x) = x³ – 2x² – 5x + 6 = 0

p(3) = (3)³ – 2(3)² – 5(3) + 6

= 27 – 2 × 9 – 5 × 3 + 6

= 27 – 18 – 15 + 6

= 33 – 33 = 0

∴ x = 3 is a root of x³ – 2x² – 5x + 6 = 0.