Question

What number should be subtracted from 2x³ – 5x² + 5x so that the resulting polynomial has 2x – 3 as a factor?

Answer 1

Let the number to be subtracted be k and the resulting polynomial be f(x), then
f(x) = 2x³ – 5x² + 5x – k
Since, 2x – 3 is a factor of f(x),
Now, converting 2x – 3 to factor theorem

`f(3/2)` = 0
⇒ 2x³ – 5x² + 5x – k = 0

⇒ `2(3/2)^3 - 5(3/2)^2 +5(3/2) -k` = 0

⇒ `2 xx (27)/(8) - 5 xx (9)/(4) + 5 xx (3)/(2) - k` = 0

⇒ `(27)/(4) - (45)/(4) + (15)/(2) - k` = 0

⇒ 27 – 45 + 30 – 4k = 0
⇒ –4k + 12 = 0

⇒ k = `(-12)/(-4)`
⇒ k = 3.