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P(x)=x^(4)-2x^(3)+kx-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=

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

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Evaluate piecewise-defined functions

Full solution

Q.

P(x)=x^4-2 x^3+k x-4

\newline

where

k

is an unknown integer.

\newline

P(x)

divided by

(x-1)

has a remainder of

0

.

\newline

What is the value of

k

?

\newline

k=

Apply Remainder Theorem: Apply the Remainder Theorem. $\newline$ The Remainder Theorem states that if a polynomial $P(x)$ is divided by $(x - c)$ and the remainder is $0$ , then $c$ is a root of the polynomial. This means $P(c) = 0$ .
Substitute $x = 1$ : Substitute $x = 1$ into $P(x)$ . $\newline$ Since the remainder is $0$ when $P(x)$ is divided by $(x - 1)$ , we substitute $x = 1$ into the polynomial $P(x)$ to find the value of $k$ . $\newline$ $P(1) = (1)^4 - 2(1)^3 + k(1) - 4$
Simplify the expression: Simplify the expression. $\newline$ $P(1) = 1 - 2 + k - 4$ $\newline$ $P(1) = k - 5$
Set $P(1)$ equal to $0$ : Set $P(1)$ equal to $0$ . Since $P(1)$ must be $0$ for the remainder to be $0$ , we set the simplified expression equal to $0$ . $k - 5 = 0$
Solve for k: Solve for k. $\newline$ Add $5$ to both sides of the equation to solve for k. $\newline$ $k = 5$

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