Problem 29 - College Algebra

Question

Divide using synthetic division. $$\frac{x^{4}-256}{x-4}$$

Step-by-Step Solution

Answer

The quotient is \(x^{3}+4x^{2}+16x+64\).

Step 1: Set Up the Synthetic Division

First, write down the coefficients of the polynomial being divided, which 'dividend' is \(x^{4}-256\). The dividend polynomial has coefficients: [1, 0, 0, 0, -256], corresponding to powers \(x^4, x^3, x^2, x^1, x^0\), respectively. The divisor is \(x-4\) and the zero is just 4.

Step 2: Perform the Synthetic Division

Bring down the first coefficient (1), multiply it by 4 (the zero of the divisor), and write the result (4) under the second coefficient (0). Add to get a new second coefficient, which is 4 (0 + 4). Repeat these steps for subsequent coefficients: multiply the new coefficient by 4, write the result under next coefficient and add. Following this rule, the sequence of new coefficients is [1, 4, 16, 64, 256].

Step 3: Write the Answer

The last number (256) is the remainder, and the other numbers are the coefficients of the quotient polynomial. The powers of the quotient polynomial start from one less than the original polynomial (here \(x^3\)). Hence, the answer is \(x^{3}+4x^{2}+16x+64+\frac{256}{x-4}\), where \(\frac{256}{x-4}\) is zero, thus solution is \(x^{3}+4x^{2}+16x+64\).