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P(x)=x^(4)-3x^(2)+kx-2
where
k is an unknown integer.

P(x) divided by
(x-2) has a remainder of 10 .
What is the value of
k ?

k=

$P(x)=x^{4}-3 x^{2}+k x-2$ $\newline$ where $k$ is an unknown integer. $\newline$ $P(x)$ divided by $(x-2)$ has a remainder of $10$ . $\newline$ What is the value of $k$ ? $\newline$ $k=$

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Evaluate recursive formulas for sequences

Full solution

Q.

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

\newline

where

k

is an unknown integer.

\newline

P(x)

divided by

(x-2)

has a remainder of

10

.

\newline

What is the value of

k

?

\newline

k=

Apply Remainder Theorem: To find the value of $k$ , we will use the Remainder Theorem, which states that if a polynomial $P(x)$ is divided by $(x - c)$ , the remainder is $P(c)$ . In this case, $c$ is $2$ because we are dividing by $(x - 2)$ , and the remainder is given as $10$ . So we need to find $P(2)$ .
Substitute $x = 2$ : We substitute $x = 2$ into the polynomial $P(x) = x^4 - 3x^2 + kx - 2$ .
$P(2) = (2)^4 - 3(2)^2 + k(2) - 2$
$P(2) = 16 - 3(4) + 2k - 2$
$P(2) = 16 - 12 + 2k - 2$
Simplify $P(2)$ : Now we simplify the expression for $P(2)$ .
$P(2) = 16 - 12 - 2 + 2k$
$P(2) = 2 + 2k$
Set $P(2) = 10$ : Since the remainder when $P(x)$ is divided by $(x - 2)$ is $10$ , we set $P(2)$ equal to $10$ . $\newline$ $2 + 2k = 10$
Solve for k: We solve for k by subtracting $2$ from both sides of the equation. $\newline$ $2k = 10 - 2$ $\newline$ $2k = 8$
Final Value of k: Now we divide both sides by $2$ to find the value of k. $\newline$ k = $\frac{8}{2}$ $\newline$ k = $4$

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