what is the remainder when you divide $(k^{5}-4k^{3}+7k)$ by $(k - 1)$?\n$-\frac{4}{k - 1}$\n$-\frac{1}{k…

what is the remainder when you divide $(k^{5}-4k^{3}+7k)$ by $(k - 1)$?\n$-\frac{4}{k - 1}$\n$-\frac{1}{k - 1}$\n$\frac{4}{k - 1}$\n$\frac{1}{k - 1}$

what is the remainder when you divide $(k^{5}-4k^{3}+7k)$ by $(k - 1)$?\n$-\frac{4}{k - 1}$\n$-\frac{1}{k - 1}$\n$\frac{4}{k - 1}$\n$\frac{1}{k - 1}$

Answer

Explanation:

Step1: Apply the Remainder Theorem

The Remainder Theorem states that when a polynomial (f(k)) is divided by ((k - a)), the remainder is (f(a)). Here, (a = 1) and (f(k)=k^{5}-4k^{3}+7k).

Step2: Substitute (k = 1) into the polynomial

Substitute (k = 1) into (f(k)): [ \begin{align*} f(1)&=(1)^{5}-4(1)^{3}+7(1)\ &=1 - 4+7\ &=4 \end{align*} ] Since the remainder is a constant (not a rational - function form as the options seem to be mis - presented. If we assume it's a mis - type and we are actually using the Remainder Theorem in the correct polynomial - division sense), when dividing a polynomial (P(k)) by ((k - 1)), the remainder (r) is a constant.

If we consider the polynomial long - division of (P(k)=k^{5}-4k^{3}+7k) by (D(k)=k - 1). We can also use the formula (P(k)=(k - 1)Q(k)+r), where (Q(k)) is the quotient polynomial and (r) is the remainder. When (k = 1), (P(1)=r)

Answer:

(4)