Divide using long division or synthetic division.
(21x^3 - 7)/(3x - 1)
Just wondering, how would you solve this problem using synthetic division?
I tried it and the quotient is not a polynomial. It's a mess with a remainder and fractional coefficients.
I got 3*[x^2 + x/3 + 1/9 -(8/9)/(3x-1)]
For a tutorial on long and synthetic division, see
http://faculty.ed.umuc.edu/~swalsh/Math%20Articles/Synthetic%20Division.html
To divide this polynomial using synthetic division, you will need to follow these steps:
Step 1: Arrange the terms of the dividend in descending order of the exponent.
The given polynomial is already in descending order, so we can move on to the next step.
Step 2: Set up the synthetic division table.
In synthetic division, we write down the coefficients of the terms in the dividend polynomial, including any missing exponents. Since there is no x^2 term, we need to include a placeholder of 0.
1 | 21 0 0 -7
Step 3: Determine the divisor.
The divisor in this case is (3x - 1). To use synthetic division, we need to write this in the form (x - k), where k is the value of x that makes the divisor equal to zero. In this case, setting 3x - 1 equal to zero and solving for x gives x = 1/3.
1/3 | 21 0 0 -7
Step 4: Perform synthetic division.
Starting with the first coefficient (21), bring it down below the line.
1/3 | 21 0 0 -7
21
Next, multiply the divisor (1/3) by the number below the line (21) and write the result below the next coefficient (0). Then, add the two numbers together.
1/3 | 21 0 0 -7
21
--------------
0
Repeat this process for each coefficient until all terms have been divided.
1/3 | 21 0 0 -7
21 7
--------------
0 7 0
The last number on the line (-7) represents the remainder. The numbers on the top line (0, 7, 0) are the coefficients of the resulting quotient polynomial.
Therefore, the quotient of (21x^3 - 7) divided by (3x - 1) is 7x^2 + 0x + 0, or simply 7x^2.