$\begin{array}{l}P(x)=x^{4}-2 x^{2}-8 \\ d(x)=x+2\end{array}$

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Find higher derivatives of rational and radical functions

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\begin{array}{l}P(x)=x^{4}-2 x^{2}-8 \\ d(x)=x+2\end{array}

Set Up Polynomial Division: First, let's set up the polynomial division, where $P(x)$ is the dividend and $d(x)$ is the divisor. $\newline$ $P(x) = x^4 - 2x^2 - 8$ $\newline$ $d(x) = x + 2$ $\newline$ We will use long division to divide $P(x)$ by $d(x)$ .
Divide First Term: Divide the first term of $P(x)$ by the first term of $d(x)$ : $x^4$ divided by $x$ gives us $x^3$ . Write $x^3$ above the division bar.
Subtract and Find Remainder: Multiply $d(x)$ by $x^3$ and write the result under $P(x)$ : $(x + 2)(x^3) = x^4 + 2x^3$ . Subtract this from $P(x)$ to find the remainder.
Divide New Remainder: Subtracting we get: $x^4 - 2x^2 - 8$ - $x^4 + 2x^3$ = (-2\)x^ $3$ - $2$ x^ $2$ - $8$ . Bring down the next term if necessary to continue the division.
Continue Division: Divide the first term of the new remainder by the first term of $d(x)$ : $-2x^3$ divided by $x$ gives us $-2x^2$ . Write $-2x^2$ above the division bar next to $x^3$ .
Multiply and Subtract: Multiply $d(x)$ by $-2x^2$ and write the result under the current remainder: $(x + 2)(-2x^2) = -2x^3 - 4x^2$ . Subtract this from the current remainder to find the new remainder.
Find New Remainder: Subtracting we get: $(-2x^3 - 2x^2 - 8) - (-2x^3 - 4x^2) = 2x^2 - 8$ .
Divide New Remainder: Divide the first term of the new remainder by the first term of $d(x)$ : $2x^2$ divided by $x$ gives us $2x$ . Write $2x$ above the division bar next to $-2x^2$ .
Multiply and Subtract: Multiply $d(x)$ by $2x$ and write the result under the current remainder: $(x + 2)(2x) = 2x^2 + 4x$ . Subtract this from the current remainder to find the new remainder.
Find New Remainder: Subtracting we get: $2x^2 - 8$ - $2x^2 + 4x$ = (-4\)x - $8$ .
Divide New Remainder: Divide the first term of the new remainder by the first term of $d(x)$ : $-4x$ divided by $x$ gives us $-4$ . Write $-4$ above the division bar next to $2x$ .
Multiply and Subtract: Multiply $d(x)$ by $-4$ and write the result under the current remainder: $(x + 2)(-4) = -4x - 8$ . Subtract this from the current remainder to find the new remainder.
Find New Remainder: Subtracting we get: $(-4x - 8) - (-4x - 8) = 0$ . We have now finished the division since the remainder is $0$ .
Finish Division: The result of the polynomial division of $P(x)$ by $d(x)$ is the quotient we wrote above the division bar: $x^3 - 2x^2 + 2x - 4$ .