Algebraic division practice worksheet featuring 24 problems for students to solve, provided by Learners Planet.

The task involves solving polynomial division problems. Each problem requires dividing a polynomial by another polynomial. We will solve one of the problems as an example, and then outline the general approach for solving these types of problems.

Example Problem: Solve \( (9p^2 - 56p - 49) \div (9p + 7) \)

#### Step 1: Set up the division
We need to divide \( 9p^2 - 56p - 49 \) by \( 9p + 7 \).

#### Step 2: Divide the leading terms
- The leading term of the dividend is \( 9p^2 \).
- The leading term of the divisor is \( 9p \).
- Divide \( 9p^2 \) by \( 9p \):
\[
\frac{9p^2}{9p} = p
\]
So, the first term of the quotient is \( p \).

#### Step 3: Multiply the entire divisor by the first term of the quotient
- Multiply \( 9p + 7 \) by \( p \):
\[
p \cdot (9p + 7) = 9p^2 + 7p
\]

#### Step 4: Subtract the result from the original dividend
- Subtract \( 9p^2 + 7p \) from \( 9p^2 - 56p - 49 \):
\[
(9p^2 - 56p - 49) - (9p^2 + 7p) = 9p^2 - 56p - 49 - 9p^2 - 7p = -63p - 49
\]

#### Step 5: Repeat the process with the new polynomial
- The new polynomial is \( -63p - 49 \).
- Divide the leading term \( -63p \) by the leading term of the divisor \( 9p \):
\[
\frac{-63p}{9p} = -7
\]
So, the next term of the quotient is \( -7 \).

#### Step 6: Multiply the entire divisor by the new term of the quotient
- Multiply \( 9p + 7 \) by \( -7 \):
\[
-7 \cdot (9p + 7) = -63p - 49
\]

#### Step 7: Subtract the result from the new polynomial
- Subtract \( -63p - 49 \) from \( -63p - 49 \):
\[
(-63p - 49) - (-63p - 49) = -63p - 49 + 63p + 49 = 0
\]

#### Final Answer
The quotient is \( p - 7 \) and the remainder is \( 0 \). Therefore, the solution is:
\[
\boxed{p - 7}
\]

General Approach for Polynomial Division

1. Set up the division: Write the dividend and divisor in standard form.
2. Divide the leading terms: Divide the leading term of the dividend by the leading term of the divisor to get the first term of the quotient.
3. Multiply and subtract: Multiply the entire divisor by the term obtained and subtract the result from the dividend.
4. Repeat: Repeat the process with the new polynomial until the degree of the remainder is less than the degree of the divisor.
5. Write the answer: The final answer is the quotient plus the remainder divided by the divisor.

Additional Notes

- If the remainder is zero, the division is exact, and the quotient is the final answer.
- If the remainder is not zero, the final answer is written as:
\[
\text{Quotient} + \frac{\text{Remainder}}{\text{Divisor}}
\]

This method can be applied to all the problems listed in the image. If you need solutions for specific problems, please let me know!