Find the remainder when the polynomial \( f(x) = 3x^4 - 2x^3 + 4x^2 - 5x + 1 \) is divided by the polynomial \( g(x) = x^2 - 3x + 2 \).
Answer
We can't continue the process because the degree of the current dividend \(10x - 7\) is less than the degree of the divisor \(x^2 - 3x + 2\). Thus, the remainder is \(10x - 7\).
Step 1 ：First, set up the polynomial division in long division format.
Step 2 ：Next, divide the leading term of the dividend (\(3x^4\)) by the leading term of the divisor (\(x^2\)), to get \(3x^2\). Write this term above the division bar.
Step 3 ：Then, multiply the divisor \(x^2 - 3x + 2\) by \(3x^2\) and subtract the result from the dividend, to get \(-x^3 + 7x^2 - 5x + 1\).
Step 4 ：Repeat the process: divide \(-x^3\) by \(x^2\) to get \(-x\), multiply the divisor by \(-x\), and subtract the result from the current dividend to get \(4x^2 - 2x + 1\).
Step 5 ：Repeat the process again: divide \(4x^2\) by \(x^2\) to get \(4\), multiply the divisor by \(4\), and subtract the result from the current dividend to get \(10x - 7\).
Step 6 ：We can't continue the process because the degree of the current dividend \(10x - 7\) is less than the degree of the divisor \(x^2 - 3x + 2\). Thus, the remainder is \(10x - 7\).
