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