(x^4-y^4)/(x-y) Synthetic Division

4 min read Jun 17, 2024

Simplifying (x⁴ - y⁴) / (x - y) using Synthetic Division

Synthetic division is a handy tool for dividing polynomials, especially when the divisor is of the form (x - a). Let's see how to apply it to simplify the expression (x⁴ - y⁴) / (x - y).

Setting up the Synthetic Division

Identify the coefficients: Our polynomial is x⁴ - y⁴. Since we are dividing by (x - y), we will use y as our "a" value. The coefficients of our polynomial are 1, 0, 0, 0, and -y⁴ (notice the zero coefficients for the missing terms).
Set up the synthetic division table: Draw a table with the coefficients of the polynomial on the top row and the "a" value (y in this case) to the left.
```
  y | 1   0   0   0   -y⁴
    |______________________
```

Performing the Synthetic Division

Bring down the first coefficient: Bring down the first coefficient (1) below the line.
```
  y | 1   0   0   0   -y⁴
    |______________________
        1 
```
Multiply and add: Multiply the "a" value (y) by the number below the line (1) and write the result (y) under the second coefficient (0). Add the two numbers together and write the result below the line.
```
  y | 1   0   0   0   -y⁴
    |______________________
        1   y 
```

Repeat the process: Repeat steps 2 and 3 for the remaining coefficients:

  y | 1   0   0   0   -y⁴
    |______________________
        1   y   y²  y³

Final step: Multiply the last number below the line (y³) by the "a" value (y) and add it to the last coefficient (-y⁴) to get zero. This means the remainder is zero.
```
  y | 1   0   0   0   -y⁴
    |______________________
        1   y   y²  y³  0
```

Interpretation of the Results

The numbers below the line (1, y, y², y³) represent the coefficients of the quotient polynomial. Since the remainder is zero, the quotient is the simplified form of the original expression.

Therefore, (x⁴ - y⁴) / (x - y) = x³ + xy² + y³.

Verification

You can verify this result by multiplying (x - y) with (x³ + xy² + y³) and confirming that the result is indeed x⁴ - y⁴.