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Chapter 3: Problem 29

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

Short Answer

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

Step by step solution

01

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.
02

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].
03

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

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