https://file.aiquickdraw.com/mathbot/file/1726026409936krevgmsw.jpg

To solve this matching problem, we'll perform polynomial division on each expression on the left-hand side and match them with the appropriate quotient on the right-hand side.

### First expression:
\[
\frac{x^2 - x - 30}{x - 6}
\]

We'll use long division here.

1. Divide the first term: \(\frac{x^2}{x} = x\).
2. Multiply \(x\) by \(x - 6\): \(x(x - 6) = x^2 - 6x\).
3. Subtract: \((x^2 - x - 30) - (x^2 - 6x) = 5x - 30\).
4. Divide the new first term: \(\frac{5x}{x} = 5\).
5. Multiply \(5\) by \(x - 6\): \(5(x - 6) = 5x - 30\).
6. Subtract: \((5x - 30) - (5x - 30) = 0\).

Thus, the quotient is:
\[
x + 5
\]

This matches with the second option on the right.

---

### Second expression:
\[
\frac{x^3 - 2x^2 - 7x - 4}{x^2 + 2x + 1}
\]

Perform polynomial long division again.

1. Divide the first term: \(\frac{x^3}{x^2} = x\).
2. Multiply \(x\) by \(x^2 + 2x + 1\): \(x(x^2 + 2x + 1) = x^3 + 2x^2 + x\).
3. Subtract: \((x^3 - 2x^2 - 7x - 4) - (x^3 + 2x^2 + x) = -4x^2 - 8x - 4\).
4. Divide the new first term: \(\frac{-4x^2}{x^2} = -4\).
5. Multiply \(-4\) by \(x^2 + 2x + 1\): \(-4(x^2 + 2x + 1) = -4x^2 - 8x - 4\).
6. Subtract: \((-4x^2 - 8x - 4) - (-4x^2 - 8x - 4) = 0\).

Thus, the quotient is:
\[
x - 4
\]

This matches with the third option on the right.

---

### Third expression:
\[
\frac{x^3 + 2x^2 - 1}{x^2 - x + 1}
\]

Perform polynomial long division again.

1. Divide the first term: \(\frac{x^3}{x^2} = x\).
2. Multiply \(x\) by \(x^2 - x + 1\): \(x(x^2 - x + 1) = x^3 - x^2 + x\).
3. Subtract: \((x^3 + 2x^2 - 1) - (x^3 - x^2 + x) = 3x^2 - x - 1\).
4. Divide the new first term: \(\frac{3x^2}{x^2} = 3\).
5. Multiply \(3\) by \(x^2 - x + 1\): \(3(x^2 - x + 1) = 3x^2 - 3x + 3\).
6. Subtract: \((3x^2 - x - 1) - (3x^2 - 3x + 3) = 2x - 4\).

Thus, the quotient is:
\[
x + 3 \, \text{R} \, 2x - 4
\]

This matches with the first option on the right.

---

### Final Matches:
1. \(\frac{x^2 - x - 30}{x - 6} \quad \text{matches} \quad x + 5\)
2. \(\frac{x^3 - 2x^2 - 7x - 4}{x^2 + 2x + 1} \quad \text{matches} \quad x - 4\)
3. \(\frac{x^3 + 2x^2 - 1}{x^2 - x + 1} \quad \text{matches} \quad x + 3 \, \text{R} \, 2x - 4\)

Let me know if you would like any further details or explanation.

Here are 5 related questions to deepen understanding:
1. How is synthetic division used in dividing polynomials?
2. Can polynomial division be applied to non-monomial divisors?
3. How does the degree of the dividend and divisor affect the quotient?
4. What are the differences between polynomial long division and synthetic division?
5. What happens if a polynomial division has a remainder of zero?

**Tip:** Always check your quotient by multiplying it with the divisor to see if you get the original dividend, helping you verify your result.

Match each expression on the left with its quotient on the right.

1. (x^2 - x - 30) ÷ (x - 6)
2. (x^3 - 2x^2 - 7x - 4) ÷ (x^2 + 2x + 1)
3. (x^3 + 2x^2 - 1) ÷ (x^2 - x + 1)

Options:
1. x + 3 R 2x - 4
2. x + 5
3. x - 4

Learn how to solve polynomial division problems by matching expressions with their correct quotients. This problem covers key algebraic concepts including polynomial division and the division algorithm for polynomials. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation