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Question
When divided by x – 3 the polynomials x3 – px2 + x + 6 and 2x3 – x2 – (p + 3) x – 6 leave the same remainder. Find the value of ‘p’.
Solution
If (x – 3) divides f(x) = x3 – px2 + x + 6, then,
Remainder = f(3) = 33 – p(3)2 + 3 + 6 = 36 – 9p
If (x – 3) divides g(x) = 2x3 – x2 – (p + 3) x – 6, then
Remainder = g(3) = 2(3)3 – (3)2 – (p + 3) (3) – 6 = 30 – 3p
Now, f(3) = g(3)
⇒ 36 – 9p = 30 – 3p
⇒ -6p = -6
⇒ p = 1
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