Questions: Perform long division given (x^3+3x^2-3x+1)/(x-2). Identify the quotient and remainder.

Transcript text: Perform long division given $\frac{x^{3}+3 x^{2}-3 x+1}{x-2}$. Identify the quotient and remainder.

Solution

Solution Steps

To perform polynomial long division, divide the leading term of the dividend by the leading term of the divisor to get the first term of the quotient. Multiply the entire divisor by this term and subtract from the dividend. Repeat the process with the new polynomial until the degree of the remainder is less than the degree of the divisor. The result will be the quotient and the remainder.

Step 1: Perform Polynomial Long Division

We start with the polynomial division of $ \frac{x^{3} + 3x^{2} - 3x + 1}{x - 2} $. The first term of the quotient is obtained by dividing the leading term of the dividend $ x^{3} $ by the leading term of the divisor $ x $, which gives us $ 1x^{2} $.

Step 2: Multiply and Subtract

Next, we multiply the entire divisor $ x - 2 $ by $ x^{2} $ to get $ x^{3} - 2x^{2} $. We then subtract this from the original polynomial: \[ (x^{3} + 3x^{2} - 3x + 1) - (x^{3} - 2x^{2}) = 5x^{2} - 3x + 1 \]

Step 3: Repeat the Process

Now, we repeat the process with the new polynomial $ 5x^{2} - 3x + 1 $. The next term of the quotient is obtained by dividing $ 5x^{2} $ by $ x $, which gives us $ 5x $. We multiply $ x - 2 $ by $ 5x $ to get $ 5x^{2} - 10x $ and subtract: \[ (5x^{2} - 3x + 1) - (5x^{2} - 10x) = 7x + 1 \]

Step 4: Final Division

We continue with $ 7x + 1 $. The next term of the quotient is $ 7 $ (since $ 7x \div x = 7 $). We multiply $ x - 2 $ by $ 7 $ to get $ 7x - 14 $ and subtract: \[ (7x + 1) - (7x - 14) = 15 \]