$P(x)=x^{4}-2 x^{3}+k x-4$ where $k$ is an unknown integer.
$P(x)$ divided by $(x-1)$ has a remainder of 0 .
What is the value of $k ?$
$$
k=
$$
$\square$

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Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Given the polynomial $P(x) = x^4 - 2x^3 + kx - 4$, we need to find the value of $k$ such that the remainder is $0$ when $P(x)$ is divided by $(x - 1)$. ‖ ‖ step 2 ⋮ To find the remainder when $P(x)$ is divided by $(x - 1)$, we can use the Remainder Theorem, which states that the remainder is $P(1)$. ‖ ‖ step 3 ⋮ Substitute $x = 1$ into the polynomial: $P(1) = 1^4 - 2 \cdot 1^3 + k \cdot 1 - 4$. ‖ ‖ step 4 ⋮ Simplify the expression: $P(1) = 1 - 2 + k - 4$. ‖ ‖ step 5 ⋮ Solve for $k$: $1 - 2 + k - 4 = 0 \Rightarrow k = 5$. ‖ ∻[1] Answer∻ ⚹ $k = 5$ ⚹ ∻Key Concept∻ ⚹Remainder Theorem⚹ ∻Explanation∻ ⚹The Remainder Theorem states that the remainder of the division of a polynomial $P(x)$ by a linear divisor $(x - c)$ is equal to $P(c)$. In this case, $c = 1$, and we find $k$ by setting $P(1) = 0$.⚹◊What is the method to determine the value of the unknown coefficient k in the polynomial function P(x)?⍭ Generate me a similar question◊

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