The value of k in the polynomial is 3.
What is remainder theorem?
It states that if you divide a polynomial P(x) by a linear divisor of the form (x - c), the remainder is equal to P(c).
Given a polynomial P(x) which is divided by (x - c) , the remainder R is given by:
P(x) = (x - c)Q(x) + R
where are Q(x) is the quotient obtained from the division, and R = P(c).
Using the remainder theorem
Given
f(x) =3x⁴ -5x³ + kx² -5x - 2
When f(x) is divide by x - 2 the remember is 8
Therefore,
When x - 2 = 0
x = 2
f(2) = 8
f(2) = 3(2)⁴ -5(2)³ + k(2)² - 5(2) - 2 = 8
Simplifying
3*16 -5*8 + k*4 - 10 - 2 = 8
48 - 40 + 4k - 10 - 2 = 8
4k - 4 = 8
4k = 8 + 4
4k = 12
k = 12/4
= 3
The value of k in the polynomial is 3.