Problem 76 - College Algebra

Question

Find the quotient of \(x^{3 n}+1\) and \(x^{n}+1\)

Step-by-Step Solution

Answer

The quotient of \(x^{3 n}+1\) and \(x^{n}+1\) is \(x^{2 n}\).

Step 1: Identify Terms for Division

We need to find the quotient when \(x^{3 n}+1\) is divided by \(x^n+1\). Here, \(x^{3 n}+1\) is the dividend and \(x^{n}+1\) is the divisor.

Step 2: Perform Division using Polynomial Division

Now, we perform the division as we would in regular polynomial division.\n Divide \(x^{3 n}\) by \(x^{n}\) to get \(x^{2 n}\). Then multiply \(x^{n}+1\) by the result, which gives \( x^{3 n} + x^{2 n}\). Subtract this from \(x^{3 n}+1\) to get \(- x^{2 n} + 1\). As the degree of \(- x^{2 n} + 1\) is less than the degree of \(x^{n}+1\), we stop here, and \(x^{2 n}\) is our quotient.

Step 3: Result

The quotient of the division \(x^{3 n}+1\) divided by \(x^{n}+1\) is \(x^{2 n}\).