Questions: Use polynomial long division to divide -x^3-4x^2-4x by x-2.

Transcript text: Use polynomial long division to divide $-x^{3}-4 x^{2}-4 x$ by $x-2$.

Solution

Solution Steps

To solve the problem of dividing the polynomial $-x^3 - 4x^2 - 4x$ by $x - 2$ using polynomial long division, we will follow these steps:

Divide the leading term of the dividend by the leading term of the divisor.
Multiply the entire divisor by the result from step 1 and subtract it from the dividend.
Repeat the process with the new polynomial obtained after subtraction until the degree of the remainder is less than the degree of the divisor.

Step 1: Set Up the Division

We are dividing the polynomial $ -x^3 - 4x^2 - 4x $ by $ x - 2 $. The dividend is represented as $ -1x^3 - 4x^2 - 4x + 0 $ and the divisor is $ 1x - 2 $.

Step 2: Perform Polynomial Long Division

Divide the leading term of the dividend $ -x^3 $ by the leading term of the divisor $ x $: \[ \frac{-x^3}{x} = -1 \]
Multiply the entire divisor $ x - 2 $ by $-1$: \[ -1(x - 2) = -x + 2 \]
Subtract this from the original polynomial: \[ (-x^3 - 4x^2 - 4x) - (-x + 2) = -x^3 - 4x^2 - 4x + x - 2 = -x^3 - 4x^2 - 3x - 2 \]
Repeat the process with the new polynomial $ -x^3 - 4x^2 - 3x - 2 $:
- Divide $ -x^3 $ by $ x $: \[ \frac{-x^3}{x} = -1 \]
- Multiply the divisor by $-1$ and subtract again.

Continuing this process, we find that the quotient is $ -1, -6, -16 $ and the remainder is $ -32 $.

Step 3: Write the Result

The result of the polynomial long division can be expressed as: \[ \frac{-x^3 - 4x^2 - 4x}{x - 2} = -1x^2 - 6x - 16 + \frac{-32}{x - 2} \]

Final Answer

The quotient is $ -1x^2 - 6x - 16 $ and the remainder is $ -32 $. Thus, the final answer is: \[ \boxed{-1x^2 - 6x - 16 \text{ with a remainder of } -32} \]

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