Two polynomials P and D are given. Use either synthetic or long division to divide
P(x) by D(x),and express the quotient
P(x)/D(x) in the form P(x)D(x) = Q(x)+
R(x)D(x)
.
P(x) = 6x^4 − 3x^3 + 29x^2, D(x) = 3x^2 + 13
Hey, I did this down below.
To divide P(x) by D(x) using synthetic division, follow these steps:
Step 1: Write the coefficients of the polynomials P(x) and D(x) in descending order of powers of x:
P(x) = 6x^4 - 3x^3 + 29x^2 + 0x + 0
D(x) = 3x^2 + 0x + 13
Step 2: Perform synthetic division by following these steps:
a) Set up the synthetic division table:
| 3 | 0 | 13
_________|________|_______|______
6 | | |
b) Bring down the first coefficient of P(x), which is 6:
| 3 | 0 | 13
_________|________|_______|______
6 | | |
|
c) Multiply the divisor (3) by the number at the bottom of the table (6), and write the result above the line:
| 3 | 0 | 13
_________|________|_______|______
6 | 18 | |
|
d) Add the numbers in the same column:
| 3 | 0 | 13
_________|________|_______|______
6 | 18 | |
|
e) Repeat steps c) and d) until all coefficients of P(x) have been processed:
| 3 | 0 | 13
_________|________|_______|______
6 | 18 | -54 | 73
|
Step 3: The numbers at the bottom of the table represent the coefficients of the quotient polynomial Q(x):
Q(x) = 18x - 54
Step 4: The remainder is the last number in the table divided by the divisor (3):
Remainder = 73 / 3 = 24
Step 5: Rewrite the division equation in the desired form:
P(x) = (Q(x) * D(x)) + Remainder
P(x) = (18x - 54) * (3x^2 + 13) + 24*D(x)
So the quotient P(x)/D(x) in the desired form is:
P(x)/D(x) = 18x^3 + 78x - 54x^2 - 234 + 24*(3x^2 + 13)