Question

use synthetic division to find the remainder.\\(\frac{-x^3 - 5x^2 - 9x - 24}{x + 4}\\)

Explanation:

Step1: Identify the root of the divisor

The divisor is \(x + 4\), so we set \(x+4 = 0\), which gives \(x=-4\). We will use \(-4\) for synthetic division. Also, write the coefficients of the dividend \(-x^{3}-5x^{2}-9x - 24\). The coefficients are \(-1\) (for \(x^{3}\)), \(-5\) (for \(x^{2}\)), \(-9\) (for \(x\)), and \(-24\) (constant term).

Step2: Set up synthetic division

Set up the synthetic division as follows:
Bring down the first coefficient (\(-1\)):

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Step3: Multiply and add

Multiply the brought - down value (\(-1\)) by \(-4\): \((-1)\times(-4)=4\). Add this to the next coefficient (\(-5\)): \(-5 + 4=-1\).

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Multiply the new value (\(-1\)) by \(-4\): \((-1)\times(-4) = 4\). Add this to the next coefficient (\(-9\)): \(-9+4=-5\).

$$
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Multiply the new value (\(-5\)) by \(-4\): \((-5)\times(-4)=20\). Add this to the last coefficient (\(-24\)): \(-24 + 20=-4\).

$$
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The last value in the synthetic division result is the remainder.

Answer:

The remainder is \(-4\)