(x^2 - 6x - 20) ÷ (x + 2)

Understand the Problem

The question is asking for the result of dividing the polynomial (x^2 - 6x - 20) by (x + 2). This involves using polynomial long division or synthetic division to find the quotient and remainder.

Answer

$$ x - 8 - \frac{4}{x + 2} $$

Steps to Solve

Set up the division We are dividing the polynomial $P(x) = x^2 - 6x - 20$ by $D(x) = x + 2$.
Divide the leading terms Divide the leading term of the polynomial by the leading term of the divisor: $$ \frac{x^2}{x} = x $$
Multiply and subtract Multiply the entire divisor $D(x)$ by the result from step 2: $$ (x)(x + 2) = x^2 + 2x $$

Now subtract this from the original polynomial: $$ (x^2 - 6x - 20) - (x^2 + 2x) $$ This simplifies to: $$ -6x - 2x - 20 = -8x - 20 $$

Repeat the process Now repeat the process with the new polynomial $-8x - 20$.

Divide the leading term: $$ \frac{-8x}{x} = -8 $$

Multiply and subtract again Multiply the divisor by the new quotient: $$ (-8)(x + 2) = -8x - 16 $$

Subtract this from $-8x - 20$: $$ (-8x - 20) - (-8x - 16) $$ This simplifies to: $$ -20 + 16 = -4 $$

Write the final result We can now express the final result of the division as: $$ \text{Quotient} = x - 8 $$ $$ \text{Remainder} = -4 $$

Thus, the result of the division is: $$ \frac{x^2 - 6x - 20}{x + 2} = x - 8 - \frac{4}{x + 2} $$

The final answer is: $$ x - 8 - \frac{4}{x + 2} $$

More Information

This result shows that when you divide the polynomial $P(x) = x^2 - 6x - 20$ by $D(x) = x + 2$, the quotient is $x - 8$ and the remainder is $-4$. This indicates that $D(x)$ is not a factor of $P(x)$.

Tips

Forgetting to subtract the multiplied results correctly after each step can lead to incorrect coefficients.
Not aligning the terms properly in polynomial long division can cause confusion.
Failing to explicitly state the final quotient and remainder can lead to incomplete answers.

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