Step-by-step Sample Problem ion to divide the two polynomials. \( -2 w-9)-8 w^{4}-34 w^{3}+5 w^{2}-10 w+36 \) in ( \( X-r \) ) form? No to multiply the divisor by to get a coefficient of 1 ? are in the dividend? 5

To divide the polynomial \( -8w^4 - 34w^3 + 5w^2 - 10w + 36 \) by \( -2w - 9 \), we will use polynomial long division. Let's break this down step by step. ### Step 1: Set Up the Division We are dividing: - **Dividend**: \( -8w^4 - 34w^3 + 5w^2 - 10w + 36 \) - **Divisor**: \( -2w - 9 \) ### Step 2: Divide the Leading Terms 1. Divide the leading term of the dividend \( -8w^4 \) by the leading term of the divisor \( -2w \): \[ \frac{-8w^4}{-2w} = 4w^3 \] This is the first term of our quotient. ### Step 3: Multiply and Subtract 2. Multiply the entire divisor \( -2w - 9 \) by \( 4w^3 \): \[ 4w^3(-2w - 9) = -8w^4 - 36w^3 \] 3. Subtract this result from the original dividend: \[ (-8w^4 - 34w^3) - (-8w^4 - 36w^3) = 2w^3 \] Now, our new polynomial is: \[ 2w^3 + 5w^2 - 10w + 36 \] ### Step 4: Repeat the Process 4. Divide the leading term \( 2w^3 \) by \( -2w \): \[ \frac{2w^3}{-2w} = -w^2 \] This is the next term of our quotient. 5. Multiply the divisor by \( -w^2 \): \[ -w^2(-2w - 9) = 2w^3 + 9w^2 \] 6. Subtract this from the current polynomial: \[ (2w^3 + 5w^2) - (2w^3 + 9w^2) = -4w^2 \] Now, our new polynomial is: \[ -4w^2 - 10w + 36 \] ### Step 5: Continue Dividing 7. Divide \( -4w^2 \) by \( -2w \): \[ \frac{-4w^2}{-2w} = 2w \] This is the next term of our quotient. 8. Multiply the divisor by \( 2w \): \[ 2w(-2w - 9) = -4w^2 - 18w \] 9. Subtract this from the current polynomial: \[ (-4w^2 - 10w) - (-4w^2 - 18w) = 8w \] Now, our new polynomial is: \[ 8w + 36 \] ### Step 6: Final Division 10. Divide \( 8w \) by \( -2w \): \[ \frac{8w}{-2w} = -4 \] This is the next term of our quotient. 11. Multiply the divisor by \( -4 \): \[ -4(-2w - 9) = 8w + 36 \] 12. Subtract this from the current polynomial: \[ (8w + 36) - (8w + 36) = 0 \] ### Conclusion The division is complete, and there is no remainder. The quotient is: \[ 4w^3 - w^2 + 2w - 4 \] ### Answer to the Questions 1. The quotient in \( X - r \) form is \( 4w^3 - w^2 + 2w - 4 \). 2. The coefficient of \( 1 \) is not needed in this case since we have already divided completely. 3. The number of terms in the dividend is \( 5 \). If you have any further questions or need additional assistance, feel free to ask!