To divide the polynomial $(r^4 + 10r^3 - 124r)$ by $(r + 5)$, we will use polynomial long division.

Step-by-Step Polynomial Long Division

1. Set up the division: $\frac{r^4 + 10r^3 + 0r^2 - 124r + 0}{r + 5}$

   Notice that we added $0r^2$ and $0$ to account for all the degrees of $r$.

2. Divide the first term of the dividend by the first term of the divisor:

   $\frac{r^4}{r} = r^3$

   This will be the first term in the quotient.

3. Multiply the entire divisor by this term and subtract:

   $(r + 5) \times r^3 = r^4 + 5r^3$

   Subtract this from the original polynomial:

   $(r^4 + 10r^3 + 0r^2 - 124r + 0) - (r^4 + 5r^3) = 5r^3 + 0r^2 - 124r$

4. Repeat the process with the new polynomial:

   Divide the new leading term by the leading term of the divisor:

   $\frac{5r^3}{r} = 5r^2$

   Multiply and subtract:

   $(r + 5) \times 5r^2 = 5r^3 + 25r^2$

   $(5r^3 + 0r^2 - 124r) - (5r^3 + 25r^2) = -25r^2 - 124r$

5. Continue the process:

   $\frac{-25r^2}{r} = -25r$

   Multiply and subtract:

   $(r + 5) \times (-25r) = -25r^2 - 125r$

   $(-25r^2 - 124r) - (-25r^2 - 125r) = r$

6. Last division step:

   $\frac{r}{r} = 1$

   Multiply and subtract:

   $(r + 5) \times 1 = r + 5$

   $(r - r - 5) = -5$

7. Conclusion:

   The quotient is $r^3 + 5r^2 - 25r + 1$ and the remainder is $-5$.

Final Result

$\frac{r^4 + 10r^3 - 124r}{r + 5} = r^3 + 5r^2 - 25r + 1 - \frac{5}{r+5}$

Would you like more details or have any questions?

Related Questions

1. How do you perform polynomial division when the divisor is a quadratic expression?
2. What is the Remainder Theorem, and how does it apply to polynomial division?
3. How can synthetic division be used to simplify polynomial division when the divisor is of the form $x - c$?
4. What are the possible types of remainders in polynomial division?
5. How do you verify the result of a polynomial division?

Tip: Always arrange the polynomial terms in descending order of power to avoid mistakes in polynomial long division.