Use the remainder theorem to findvP(-2) for P(x)=2x^4+4x^3-5x^2+4.
Specifically, give the quotient and the remainder for the associated division and the value of P(-2).
Quotient=
Remainder=
P(-2)=
X = -2,
X + 2 = 0.
Using synthetic division, divide the given Eq by X + 2 and get:
P(-2) = 2X^3 - 5X + 10, R = -16.
To use the remainder theorem to find P(-2) for the polynomial P(x)=2x^4+4x^3-5x^2+4, we need to divide the polynomial by the factor (x + 2) since we're looking for the value when x = -2.
1. Start by setting up long division with the polynomial P(x) divided by (x + 2). Write the polynomial in descending order:
2x^4 + 4x^3 - 5x^2 + 4 ÷ (x + 2)
2. Begin dividing the first term of the polynomial (2x^4) by the first term of (x + 2), which is x. The result is 2x^3. Write this above the division line.
2x^3
x + 2 | 2x^4 + 4x^3 - 5x^2 + 4
3. Multiply (x + 2) by 2x^3, yielding 2x^4 + 4x^3. Write this below the polynomial.
2x^3
x + 2 | 2x^4 + 4x^3 - 5x^2 + 4
- (2x^4 + 4x^3)
4. Subtract the obtained term (2x^4 + 4x^3) from the original polynomial, canceling out the like terms. The result is -8x^3 - 5x^2 + 4.
2x^3 - x^2
x + 2 | 2x^4 + 4x^3 - 5x^2 + 4
- (2x^4 + 4x^3)
-------------
- 5x^3 - 5x^2
5. Bring down the next term, which is -5x^2.
2x^3 - x^2
x + 2 | 2x^4 + 4x^3 - 5x^2 + 4
- (2x^4 + 4x^3)
-------------
- 5x^3 - 5x^2
- 5x^2
6. Divide -5x^3 by (x + 2), which results in -5x. Write this above the division line.
2x^3 - x^2 - 5x
x + 2 | 2x^4 + 4x^3 - 5x^2 + 4
- (2x^4 + 4x^3)
-------------
- 5x^3 - 5x^2
- 5x^2
-------
0
7. Multiply (x + 2) by -5x, yielding -5x^2 - 10x. Write this below the polynomial.
2x^3 - x^2 - 5x
x + 2 | 2x^4 + 4x^3 - 5x^2 + 4
- (2x^4 + 4x^3)
-------------
- 5x^3 - 5x^2
- 5x^2
-------
- 10x
8. Subtract the obtained term (-5x^2 - 10x) from the previous result, canceling out the like terms. The result is 5x^2 - 10x + 4.
2x^3 - x^2 - 5x
x + 2 | 2x^4 + 4x^3 - 5x^2 + 4
- (2x^4 + 4x^3)
-------------
- 5x^3 - 5x^2
- 5x^2
-------
- 10x
- 10x + 4
9. We have now reached the end of the division. The quotient is 2x^3 - x^2 - 5x, and the remainder is -10x + 4.
Therefore,
Quotient = 2x^3 - x^2 - 5x
Remainder = -10x + 4
To find P(-2), substitute -2 for x in the original polynomial:
P(-2) = 2(-2)^4 + 4(-2)^3 - 5(-2)^2 + 4
Simplifying:
P(-2) = 2(16) + 4(-8) - 5(4) + 4
P(-2) = 32 - 32 - 20 + 4
P(-2) = -16 - 16
P(-2) = -32
Therefore,
Quotient = 2x^3 - x^2 - 5x
Remainder = -10x + 4
P(-2) = -32