Divide using long division or synthetic division.
(21x^3 - 7)/(3x - 1)
Just wondering, how would you solve this problem using synthetic division?
To solve this problem using synthetic division, we follow these steps:
Step 1: Write the exponent of the variable in descending order.
The dividend is 21x^3 - 7, and the divisor is 3x - 1.
Step 2: Set up the synthetic division table.
We write the coefficients of the dividend, in this case, 21, 0, 0, -7, and the constant term of the divisor, which is -1, as shown below:
|21 0 0 -7
-1 |
Step 3: Bring down the first coefficient.
Bring down the first coefficient, which is 21, in the leftmost column of the synthetic division table:
|21 0 0 -7
-1 |
21
Step 4: Multiply the divisor by the number just brought down.
Multiply -1 (divisor's constant term) by 21 (number brought down). Write the result below the next coefficient:
|21 0 0 -7
-1 | 1
21
-21
Step 5: Add the product to the next coefficient.
Add the product, -21, to the next coefficient in the dividend, which is 0, and write the result below the next coefficient:
|21 0 0 -7
-1 | 1
21
-21
----
0
Step 6: Repeat steps 4 and 5 until all coefficients have been processed.
Multiply -1 (divisor's constant term) by the sum of -21 and 0, which is -21. Write the result below the next coefficient:
|21 0 0 -7
-1 | 1 -21
21
-21
----
0
Bring down the next coefficient, which is 0:
|21 0 0 -7
-1 | 1 -21
21
-21
----
0
0
Multiply -1 (divisor's constant term) by the sum of 0 and 0, which is 0. Write the result below the next coefficient:
|21 0 0 -7
-1 | 1 -21 0
21
-21
----
0
0
0
Step 7: Read the final row of the synthetic division table.
The final row of the synthetic division table represents the coefficients of the resulting quotient. In this case, it is 1, -21, and 0, respectively.
Therefore, the quotient of (21x^3 - 7) / (3x - 1) using synthetic division is 1x^2 - 21x + 0, or simply x^2 - 21x.
To solve this problem using synthetic division, we need to follow these steps:
Step 1: Arrange the terms of the given polynomial in descending order of powers.
21x^3 - 7
Step 2: Identify the divisor and set it inside the synthetic division box.
Divisor: 3x - 1
_____________
3x - 1 | 21x^3 + 0x^2 + 0x - 7
Step 3: Bring down the coefficient of the highest power of x (in this case, 21x^3) and write it below the division line.
_____________
3x - 1 | 21x^3 + 0x^2 + 0x - 7
21x^3
Step 4: Multiply the divisor (3x - 1) by the number below the division line (21x^3).
Write the result below the next term.
_____________
3x - 1 | 21x^3 + 0x^2 + 0x - 7
21x^3
-
63x^2
Step 5: Repeat the process with the new polynomial.
_____________
3x - 1 | 21x^3 + 0x^2 + 0x - 7
21x^3
-
63x^2
- - - - - - - - - - - - - - - - - - - - - - - - -
63x^2 + 0x
Step 6: Continue the process until you reach the last term.
_____________
3x - 1 | 21x^3 + 0x^2 + 0x - 7
21x^3
-
63x^2
- - - - - - - - - - - - - - - - - - - - - - - - -
63x^2 + 0x
63x^2 - 21x
- - - - - - - - - - - - - - - - - - - - - - - - -
63x^2 + 0x
63x^2 - 21x
- - - - - - - - - - - - - - - -
21x - 7
Step 7: The last number below the division line (21x - 7) is the remainder.
Step 8: The result of the synthetic division is the quotient plus the remainder.
Therefore, (21x^3 - 7)/(3x - 1) equals 7x^2 - 21x + 21 + (21x - 7)/(3x - 1)