Use long division to divide the polynomial 24x4−24x3−18x2

Question

Use long division to divide the polynomial  24x4−24x3−18x2
  by  4x3+2x2
 . Write your answer in standard form.(1 point)

GPT-4o mini · Answer

To divide the polynomial $ 24x^4 - 24x^3 - 18x^2 $ by $ 4x^3 + 2x^2 $ using long division, follow these steps: 1. **Set up the division:** Write $ 24x^4 - 24x^3 - 18x^2 $ under the long division symbol and $ 4x^3 + 2x^2 $ outside. 2. **Divide the leading term:** Divide the leading term of the dividend $ 24x^4 $ by the leading term of the divisor $ 4x^3 $: $$ \frac{24x^4}{4x^3} = 6x. $$ 3. **Multiply and subtract:** Multiply $ 6x $ by the entire divisor $ 4x^3 + 2x^2 $: $$ 6x(4x^3 + 2x^2) = 24x^4 + 12x^3. $$ Now, subtract this from the original polynomial: $$ (24x^4 - 24x^3 - 18x^2) - (24x^4 + 12x^3) = -24x^3 - 12x^3 - 18x^2 = -36x^3 - 18x^2. $$ 4. **Repeat the process:** Now take the new polynomial $-36x^3 - 18x^2$ and repeat the division. Divide the leading term $-36x^3$ by $4x^3$: $$ \frac{-36x^3}{4x^3} = -9. $$ 5. **Multiply and subtract again:** Multiply $-9$ by the entire divisor $4x^3 + 2x^2$: $$ -9(4x^3 + 2x^2) = -36x^3 - 18x^2. $$ Subtract this result from the previous polynomial: $$ (-36x^3 - 18x^2) - (-36x^3 - 18x^2) = 0. $$ 6. **Final result:** Since the remainder is $0$, the division is complete. The quotient is: $$ 6x - 9. $$ Thus, the result of dividing $ 24x^4 - 24x^3 - 18x^2 $ by $ 4x^3 + 2x^2 $ is $$ \boxed{6x - 9}. $$