12. Use the remainder theorem to find P (-2) for P(x) =x^3+2x^2-x-7. Specifically, give the quotient and the remainder for the associated division and the value of P (-2).
Quotient =? Remainder =? P (-2) =?
P(-2) is the remainder when P(x) is divided by (x+2)
P(-2) = -5, so
P(x) = (x+2)(x^2-1) - 5
Find quotient and remainder using long division
P(x)=X^2-6x-8 ,d(x)=-4
To use the remainder theorem and find P(-2) for P(x) = x^3 + 2x^2 - x - 7, we need to perform polynomial division.
We divide P(x) by (x - a), where a is the value for which we want to find the remainder.
In this case, a = -2, because we are finding P(-2).
Let's perform the division:
___________________________
(x - (-2) | x^3 + 2x^2 - x - 7
We start by dividing x^3 by x, which gives us x^2.
Then, we multiply x^2 by (x - (-2)), which gives us x^3 - 2x^2.
x^2
___________________________
(x - (-2) | x^3 + 2x^2 - x - 7
Subtracting x^3 - 2x^2 from x^3 + 2x^2 gives us 4x^2.
We bring down the next term, which is -x.
x^2 + 4x
___________________________
(x - (-2) | x^3 + 2x^2 - x - 7
- (x^3 - 2x^2)
Subtracting -x from 4x^2 gives us 5x^2.
We multiply 5x^2 by (x - (-2)), which gives us 5x^3 - 10x^2.
x^2 + 4x + 5
___________________________
(x - (-2) | x^3 + 2x^2 - x - 7
- (x^3 - 2x^2)
________________
5x^2 - x
Subtracting 5x^2 - x from 5x^2 gives us x.
We bring down the constant term, which is -7.
x^2 + 4x + 5 + x
___________________________
(x - (-2) | x^3 + 2x^2 - x - 7
- (x^3 - 2x^2)
________________
5x^2 - x
- (5x^2 -10x)
________________
9x - 7
The remainder in the division is 9x - 7.
Therefore, the quotient is x^2 + 4x + 5 and the remainder is 9x - 7.
To find P(-2), we substitute -2 into the quotient:
P(-2) = (-2)^2 + 4(-2) + 5
= 4 - 8 + 5
= 1
Therefore, P(-2) = 1.
To use the remainder theorem to find P(-2), we need to apply synthetic division using -2 as the divisor.
First, let's list the coefficients of the polynomial P(x) = x^3 + 2x^2 - x - 7.
Coefficients: [1, 2, -1, -7]
Now, we can set up the synthetic division as follows:
-2 | 1 2 -1 -7
|_______
To perform the synthetic division, we bring down the first coefficient, which is 1, and then multiply it by -2:
-2 | 1 2 -1 -7
|_______
-2
Next, we add the result to the coefficient in the second position:
-2 | 1 2 -1 -7
|_______
-2
_______
0
We repeat the process by multiplying -2 by 0 and then adding it to the next coefficient:
-2 | 1 2 -1 -7
|_______
-2
_______
0 0
Finally, we multiply -2 by 0 and add it to the last coefficient:
-2 | 1 2 -1 -7
|_______
-2
_______
0 0 2
The result of the synthetic division gives us a remainder of 2, and since the remainder is non-zero, it means that -2 is not a root of the polynomial.
Now, let's find the quotient. The quotient is represented by the coefficients above the line in the synthetic division:
Quotient = 1 0 2
Therefore, the quotient is given by Q(x) = x^2 + 2.
To find P(-2) using the remainder theorem, we substitute -2 into the polynomial P(x) = x^3 + 2x^2 - x - 7:
P(-2) = (-2)^3 + 2(-2)^2 - (-2) - 7
= -8 + 8 + 2 - 7
= -5
Therefore, the value of P(-2) is -5.
Quotient = 1 0 2
Remainder = 2
P(-2) = -5