Given P(x) is divided by x-3 and the quotient is x^2 + 3x - 5.
Then we will write:
P(x) / (x-3) = x^2 + 3x - 5
==> P(x) = (x-3)(x^2 + 3x - 5) + R where R is the remainder.
Given that P(3) = 2
==> P(3) = 0 + R = 2
==> R = 2
==> P(x) = (x-3)(x^2 + 3x - 5) + 2
Now we will open the brackets.
==> P(x) = ( x^3 +3x^2 - 5x - 3x^2 - 9x + 15 + 2
Now we will combine like terms.
==> P(x) = x^3 -14x + 17
We have P(3) = 2 and when P(x) is divided by x-3 the quotient is x^2 + 3x - 5.
Let the remainder when P(x) is divided by x-3 be R.
So P(x) / (x - 3) = x^2 + 3x - 5 + R
=> P(x) = (x^2 + 3x - 5 )(x - 3) + R
=> P(x) = x^3 + 3x^2 - 5x - 3x^2 - 9x + 15 + R
=> P(x) = x^3 - 14x + 15 + R
Now P(3) = 2
=> 3^3 - 14*3 + 15 + R = 2
=> 27 - 42 + 15 + R = 2
=> R = 2
Therefore P(x) = x^3 - 14x + 17
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