f 2x3 – 9x2 + 4x – 7 is divided by x – 3 to give a quotient of 2x2 – 3x – 5 and
a remainder of –22 , then which of the following is true?
Question 1 options:

a) 

(x – 3)(2x2 – 3x – 5) = 22

b) 

2x3 – 9x2 + 4x – 7 = (x – 3)(2x2 – 3x – 5) – 22

c) 

2x3 – 9x2 + 4x – 7 = (x – 3)(2x2 – 3x – 5) + 22

d) 

(x – 3)(2x2 – 3x – 5) = –22

To determine the correct statement based on the division of the polynomial \(2x^3 - 9x^2 + 4x - 7\) by \(x - 3\), we can use the polynomial division theorem, which states that:

\[
\text{Dividend} = \text{Divisor} \cdot \text{Quotient} + \text{Remainder}
\]

Given:

- Dividend: \(2x^3 - 9x^2 + 4x - 7\)
- Divisor: \(x - 3\)
- Quotient: \(2x^2 - 3x - 5\)
- Remainder: \(-22\)

Applying this to the theorem:

\[
2x^3 - 9x^2 + 4x - 7 = (x - 3)(2x^2 - 3x - 5) - 22
\]

Thus, the correct option is **b)**:

\[
2x^3 - 9x^2 + 4x - 7 = (x - 3)(2x^2 - 3x - 5) - 22
\]

Would you like further details or have any questions? Here are some related questions you might consider:

1. What is the process for polynomial long division?
2. How do you verify the result of polynomial division?
3. What is the significance of the remainder in polynomial division?
4. How would the problem change if the remainder were positive?
5. Can you explain the Remainder Theorem and its application?

**Tip:** Always double-check your calculations by substituting values to verify that both sides of the equation match.

f 2x3 – 9x2 + 4x – 7 is divided by x – 3 to give a quotient of 2x2 – 3x – 5 and a remainder of –22, then which of the following is true?

Explore the solution to the polynomial division problem where f(2x^3 - 9x^2 + 4x - 7) is divided by (x - 3) with a specific quotient and remainder. Understand key algebra concepts and the polynomial division theorem suitable for students in Grades 9-12.

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