# 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**

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**

Since the quotient is of 2nd order and the divisor is of 1st order, then the order of the polynomial P(x) is :

P(x)'s order = quotient's order + divisor's order

P(x)'s order = 2 + 1

P(x)'s order = 3 order

We'll write the rule of division with reminder:

P(x) = quotient *Divisor + remainder

We know that if P(3) = 2, then the reminder is R = 2. We could write P(x) as:

P(x)= (x^2 +3x-5)(x-3) + 2

We'll remove the brackets:

P(x)= x^3 - 3x^2 + 3x^2 - 9x - 5x + 15 + 2

We'll eliminate and combine like terms:

**P(x) = x^3 - 14x + 17**