Two polynomials P and D are given. Use either synthetic or long division to divide P(x) by D(x), and express P in the form P(x) = D(x) · Q(x) + R(x).
P(x) = 4x3 + 5x + 9, D(x) = 2x + 1
P(x) =(2x + 1)*()+
To divide P(x) by D(x) using long division, follow these steps:
Step 1: Arrange the polynomials in descending order of powers of x:
P(x) = 4x^3 + 0x^2 + 5x + 9
D(x) = 2x + 1
Step 2: Divide the highest power of x in P(x) by the highest power of x in D(x) to obtain the first term of the quotient:
4x^3 divided by 2x = 2x^2
Step 3: Multiply D(x) by 2x^2 to get a temporary result:
2x^2 * (2x + 1) = 4x^3 + 2x^2
Step 4: Subtract the temporary result from P(x):
(4x^3 + 0x^2 + 5x + 9) - (4x^3 + 2x^2) = -2x^2 + 5x + 9
Step 5: Bring down the next term of P(x) to continue the division:
-2x^2 + 5x + 9
Step 6: Repeat steps 2 - 5 until the degree of the remainder is less than the degree of D(x).
Next, divide (-2x^2 + 5x + 9) by 2x:
Step 2: -2x^2 divided by 2x = -x
Step 3: Multiply D(x) by -x to get a temporary result:
- x * (2x + 1) = -2x^2 - x
Step 4: Subtract the temporary result from (-2x^2 + 5x + 9):
(-2x^2 + 5x + 9) - (-2x^2 - x) = 6x + 9
Step 5: Bring down the next term of P(x) to continue the division:
6x + 9
Step 6: Repeat steps 2 - 5 until the degree of the remainder is less than the degree of D(x).
Next, divide (6x + 9) by 2x:
Step 2: 6x divided by 2x = 3
Step 3: Multiply D(x) by 3 to get a temporary result:
3 * (2x + 1) = 6x + 3
Step 4: Subtract the temporary result from (6x + 9):
(6x + 9) - (6x + 3) = 6
Step 5: There are no more terms remaining in P(x) to bring down.
So, the final result of the long division is:
P(x) = (2x^2 - x + 3) * D(x) + 6
Therefore, P(x) = (2x^2 - x + 3) * (2x + 1) + 6