Find P(x)if P(3)=2 and if P(x) is divided by x-3 the quotient is x^2+3x-5

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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

 

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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

Approved by eNotes Editorial Team
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