Find the remainder when f(x) is divided by (x - k)
f(x) = 5x⁴ + 8x³ + 4x² - 5x + 67; k = 2

Solution:

Given a polynomial f(x) = 5x⁴ + 8x³ + 4x² - 5x + 67. The divisor (x - k) is also a linear polynomial.

Thus we apply the remainder theorem to find the remainder

When f(x) is divided by (x-a), then the remainder is f(a).

Thus here f(x) = 5x⁴ + 8x³ + 4x² - 5x + 67 is divided by (x - k)

⇒ x = k where value of k = 2 (given)

f(k = 3) = 5(2)⁴ + 8(2)³ + 4(2)² - 5(2) + 67

f(k) = 80 + 64 + 16 - 10 + 67

f(k) = 217

Find the remainder when f(x) is divided by (x - k)
f(x) = 5x⁴ + 8x³ + 4x² - 5x + 67; k = 2

Summary:

Therefore, the remainder when f(x) = 5x⁴ + 8x³ + 4x² - 5x + 67 is divided by (x - k); k = 2 is 217.