(30n³ + 66n² - 2n - 52) + (10n - 8)\nan² + bn + c + \\frac{d}{5n - 4}\na = \nb = \nc = \nd =

(30n³ + 66n² - 2n - 52) + (10n - 8)\nan² + bn + c + \\frac{d}{5n - 4}\na = \nb = \nc = \nd =

(30n³ + 66n² - 2n - 52) + (10n - 8)\nan² + bn + c + \\frac{d}{5n - 4}\na = \nb = \nc = \nd =

Answer

Explanation:

Step1: Combine like - terms

First, we have ((30n^{3}+66n^{2}-2n - 52)+(10n - 8)=30n^{3}+66n^{2}+(-2n + 10n)+(-52 - 8)=30n^{3}+66n^{2}+8n-60)

Step2: Perform polynomial long - division

We divide (30n^{3}+66n^{2}+8n - 60) by (5n-4). The formula for polynomial long - division is (\frac{f(n)}{g(n)}=q(n)+\frac{r(n)}{g(n)}), where (f(n)=30n^{3}+66n^{2}+8n - 60), (g(n)=5n - 4)

  1. Divide the leading term of (f(n)) by the leading term of (g(n)): (\frac{30n^{3}}{5n}=6n^{2})
    • Multiply (g(n)) by (6n^{2}): (6n^{2}(5n - 4)=30n^{3}-24n^{2})
    • Subtract from (f(n)): ((30n^{3}+66n^{2}+8n - 60)-(30n^{3}-24n^{2})=90n^{2}+8n - 60)
  2. Divide the leading term of the new polynomial (90n^{2}+8n - 60) by the leading term of (g(n)): (\frac{90n^{2}}{5n}=18n)
    • Multiply (g(n)) by (18n): (18n(5n - 4)=90n^{2}-72n)
    • Subtract from (90n^{2}+8n - 60): ((90n^{2}+8n - 60)-(90n^{2}-72n)=80n - 60)
  3. Divide the leading term of the new polynomial (80n - 60) by the leading term of (g(n)): (\frac{80n}{5n}=16)
    • Multiply (g(n)) by (16): (16(5n - 4)=80n - 64)
    • Subtract from (80n - 60): ((80n - 60)-(80n - 64)=4)

So, (\frac{30n^{3}+66n^{2}+8n - 60}{5n - 4}=6n^{2}+18n + 16+\frac{4}{5n - 4})

Answer:

(A = 6), (B = 18), (C = 16), (D = 4)