Next question \( \rightrightarrows \) Get a similar question You can retry this question below the quotient and remainder using long division for: \( \frac{2 x^{3}-4 x^{2}+7 x-4}{2 x^{2}+5} \) quotient is \( x-2 \) remainder is \( 3 x-4 \) estion Help: Video Read Written Example

Let's verify your long division of the polynomial \( \frac{2x^{3} - 4x^{2} + 7x - 4}{2x^{2} + 5} \). ### Step-by-Step Long Division 1. **Divide the leading terms:** \[ \frac{2x^3}{2x^2} = x \] **Quotient so far:** \( x \) 2. **Multiply the entire divisor by \( x \):** \[ x \times (2x^2 + 5) = 2x^3 + 5x \] 3. **Subtract this from the original polynomial:** \[ (2x^3 - 4x^2 + 7x - 4) - (2x^3 + 5x) = -4x^2 + 2x - 4 \] 4. **Divide the new leading term by the leading term of the divisor:** \[ \frac{-4x^2}{2x^2} = -2 \] **Quotient so far:** \( x - 2 \) 5. **Multiply the entire divisor by \( -2 \):** \[ -2 \times (2x^2 + 5) = -4x^2 - 10 \] 6. **Subtract this from the current polynomial:** \[ (-4x^2 + 2x - 4) - (-4x^2 - 10) = 2x + 6 \] ### Final Result - **Quotient:** \( x - 2 \) - **Remainder:** \( 2x + 6 \) ### Correction to Your Answer You correctly identified the **quotient** as \( x - 2 \). However, the **remainder** is \( 2x + 6 \), not \( 3x - 4 \). ### Similar Practice Problem **Divide the polynomial \( 3x^{4} - 2x^{3} + x - 5 \) by \( 3x^{2} + 4 \). Find the quotient and remainder using long division.** **Solution Steps:** 1. Divide the leading terms. 2. Multiply and subtract. 3. Repeat until the degree of the remainder is less than the degree of the divisor. 4. State the final quotient and remainder. Feel free to attempt this problem, and let me know if you need further assistance or a detailed solution!