Without actual division, show that (x3 – 3x2 – 13x + 15) is exactly divisible by (x2 + 2x - 3).
Let’s find the roots of the equation (x2 + 2x - 3)
⇒ x2 + 3x – x – 3 = 0
⇒ x(x + 3) – 1(x + 3) = 0
⸫ (x + 3) (x – 1)
Hence, if (x + 3) and (x – 1) satisfies the equation x3 – 3x2 – 13x + 15 = 0, then (x3 – 3x2 – 13x + 15) will be exactly divisible by (x2 + 2x - 3).
For x = -3,
⇒ (-3)3– 3(-3)2– 13(-3) + 15
⇒ -27 – 27 + 39 + 15 = 0
For x = 1,
⇒ 13 – 3(1)2– 13(1) + 15
⇒ 1 – 3 – 13 + 15 = 0
Hence proved.