9709 P32 - Mar 2023 - Q3
1448
Find the quotient and remainder when \(8x^3 + 4x^2 + 2x + 7\) is divided by \(4x^2 + 1\).
Solution
To divide \(8x^3 + 4x^2 + 2x + 7\) by \(4x^2 + 1\), we perform polynomial long division.
1. Divide the leading term \(8x^3\) by \(4x^2\) to get \(2x\).
2. Multiply \(2x\) by \(4x^2 + 1\) to get \(8x^3 + 2x\).
3. Subtract \(8x^3 + 2x\) from \(8x^3 + 4x^2 + 2x + 7\) to get \(4x^2 + 7\).
4. Divide \(4x^2\) by \(4x^2\) to get \(1\).
5. Multiply \(1\) by \(4x^2 + 1\) to get \(4x^2 + 1\).
6. Subtract \(4x^2 + 1\) from \(4x^2 + 7\) to get \(6\).
The quotient is \(2x + 1\) and the remainder is \(6\).
