⚡ EXAM TIP: Looking for the Perfect Answer to Write in Your Exam?

Final Answer:

Quotient q(x) = x – 3

Remainder r(x) = 7x – 9

p(x) = g(x) × q(x) + r(x)

where degree of r(x) < degree of g(x)

Given:

Dividend: p(x) = x³ – 3x² + 5x – 3

Divisor: g(x) = x² – 2

x³ ÷ x² = x

x × (x² – 2) = x³ – 2x

(x³ – 3x² + 5x – 3) – (x³ – 2x)

= x³ – 3x² + 5x – 3 – x³ + 2x

= -3x² + 7x – 3

-3x² ÷ x² = -3

-3 × (x² – 2) = -3x² + 6

(-3x² + 7x – 3) – (-3x² + 6)

= -3x² + 7x – 3 + 3x² – 6

= 7x – 9

✅ FINAL ANSWER

p(x) = g(x) × q(x) + r(x)

x³ – 3x² + 5x – 3 = (x² – 2)(ax + b) + (cx + d)

= ax³ + bx² – 2ax – 2b + cx + d

= ax³ + bx² + (-2a + c)x + (-2b + d)

💡 Note: This method is more theoretical and time-consuming. For exams, stick to the long division method shown above!

p(x) = g(x) × q(x) + r(x)

Right Hand Side (RHS):

= g(x) × q(x) + r(x)

= (x² – 2) × (x – 3) + (7x – 9)

= x²(x – 3) – 2(x – 3) + 7x – 9

= x³ – 3x² – 2x + 6 + 7x – 9

= x³ – 3x² + 5x – 3

= p(x) = Left Hand Side (LHS) ✓

✅ Verification Successful! Our answer is correct.

                 x - 3           ← Quotient
       ________________________
x² - 2 | x³ - 3x² + 5x - 3
         x³  + 0x² - 2x         ← x(x² - 2)
         ___________________
            - 3x² + 7x - 3
            - 3x² + 0x + 6      ← -3(x² - 2)
            _______________
                    7x - 9      ← Remainder
    

❌ Mistake 1: Forgetting zero coefficients

When g(x) = x² – 2, students often forget that there’s a missing ‘x’ term. Always write it as x² + 0x – 2 to avoid confusion during subtraction.

❌ Mistake 2: Sign errors during subtraction

When subtracting (-3x² + 6), many students forget to change signs. Remember: subtracting a negative means adding!
Wrong: -3x² + 7x – 3 – (-3x² + 6) = 7x – 9 ❌
Right: -3x² + 7x – 3 + 3x² – 6 = 7x – 9 ✓

❌ Mistake 3: Not checking the degree of remainder

The remainder’s degree must be less than the divisor’s degree. If your remainder is x² + something, you need to continue dividing!

❌ Mistake 4: Skipping verification

In a 3-mark question, verification can earn you that crucial extra mark. Always verify using p(x) = g(x) × q(x) + r(x).

❌ Mistake 5: Poor presentation

Write your division neatly with proper alignment. Messy work leads to calculation errors and lost marks for presentation.

💡 Quick Tip: For this question, since deg(g) = 2 and deg(p) = 3, the quotient will have degree 3 – 2 = 1, and remainder will have degree < 2 (i.e., degree 1 or 0).

💡 Trick 1: “DMSB” Method

Remember the steps: Divide, Multiply, Subtract, Bring down (if needed). Repeat until done!

💡 Trick 2: Degree Check

Quotient degree = Dividend degree – Divisor degree
In our case: 3 – 2 = 1, so quotient must be degree 1 (linear). This helps you know when to stop!

💡 Trick 3: Sign Management

When subtracting, write it as “adding the opposite”. This reduces sign errors significantly!
Instead of: (a – b) – (c – d)
Think: (a – b) + (-c + d)

💡 Trick 4: Zero Placeholder

Always write missing terms with zero coefficients. For x² – 2, write x² + 0x – 2. This keeps your columns aligned!

💡 Trick 5: Verification Shortcut

To quickly verify, check if the degrees match: deg(g × q) + deg(r) should equal deg(p).
Here: deg(2 + 1) = 3 ✓ and deg(1) = 1 < 2 ✓

From Exercise 2.2:

• Question 1 – Basic polynomial division with remainder
• Question 2 – Division with linear divisor
• Question 3 – Higher degree polynomial division
• Question 4 – Similar to this question
• Question 5 – Current question (you are here)

Related Concepts from Other Exercises:

• Exercise 2.1 – Graphical representation of polynomials
• Exercise 2.3 – Relationship between zeros and coefficients
• Exercise 2.4 – Word problems on polynomials

💡 Pro Tip: Practice all questions in Exercise 2.2 sequentially. Each question builds on the previous one, helping you master polynomial division step by step!

🏗️ 1. Engineering & Architecture

Engineers use polynomial division to simplify complex equations when designing structures, calculating stress distribution, and optimizing material usage. For example, when designing a curved bridge, the load distribution follows polynomial patterns.

💻 2. Computer Graphics & Animation

Polynomial division is used in creating smooth curves and animations. When you watch a movie with CGI, polynomial equations help create realistic motion and shapes. The division algorithm helps break down complex curves into simpler components.

📊 3. Economics & Finance

Financial analysts use polynomial division to model economic trends, predict market behavior, and calculate depreciation. For instance, when calculating compound interest over time with varying rates, polynomial division simplifies the calculations.

🔬 4. Physics & Chemistry

Scientists use polynomial division when solving differential equations that describe motion, chemical reactions, and energy transformations. For example, calculating projectile motion with air resistance involves polynomial division.

📱 5. Signal Processing & Telecommunications

Your smartphone uses polynomial division in signal processing algorithms to compress data, reduce noise, and improve call quality. Every time you stream a video, polynomial mathematics is working behind the scenes!

📊 Marking Breakdown (3 Marks Total):

⏱️ Time Management:

Recommended time: 4-5 minutes

• Reading & understanding: 30 seconds
• Setting up division: 30 seconds
• Performing division: 2 minutes
• Verification: 1 minute
• Writing final answer: 30 seconds
• Review: 30 seconds

✍️ How to Write the Perfect Answer:

1. Start with “Given”: Write p(x) and g(x) clearly
2. State what to find: “To find: Quotient q(x) and Remainder r(x)”
3. Show all steps: Don’t skip any calculation steps
4. Use proper notation: Write polynomials in descending order
5. Box the answer: Make your final answer stand out
6. Verify: Always verify using the Division Algorithm
7. Conclude: Write “Hence, q(x) = x – 3 and r(x) = 7x – 9”

⚠️ Examiner’s Note: Students often lose marks by not showing intermediate steps or skipping verification. Even if your final answer is correct, you may lose 0.5-1 mark for incomplete working!

💡 Recommendation: For this question and most CBSE board exam questions, Long Division Method is your best choice. It’s systematic, easy to follow, and examiners can clearly see your working!

Q1: Which formula should I use for polynomial division?

Answer: Use the Division Algorithm: p(x) = g(x) × q(x) + r(x), where p(x) is dividend, g(x) is divisor, q(x) is quotient, and r(x) is remainder. The degree of r(x) must be less than the degree of g(x).

Q2: What is the easiest method to divide polynomials?

Answer: Polynomial long division is the most systematic method. Arrange both polynomials in descending order of powers, divide the leading terms, multiply back, subtract, and repeat until the remainder’s degree is less than the divisor’s degree.

Q3: How do I verify my polynomial division answer?

Answer: Verify using: p(x) = g(x) × q(x) + r(x). Multiply the divisor by quotient, add the remainder, and check if you get back the original dividend polynomial. If LHS = RHS, your answer is correct!

Q4: What are common mistakes in polynomial division?

Answer: Common mistakes include: (1) Forgetting to write terms with zero coefficients, (2) Sign errors during subtraction, (3) Not arranging in descending order, (4) Stopping division too early or too late, and (5) Not verifying the answer.

Q5: Can the remainder be of higher degree than divisor?

Answer: No, the remainder must always have a degree less than the divisor. If your remainder has equal or higher degree, continue the division process. This is a fundamental rule of the Division Algorithm.

Q6: How much time should I spend on this question in exam?

Answer: For a 3-mark question on polynomial division, allocate 4-5 minutes. This includes performing division (2 min), verification (1 min), and writing the final answer clearly (30 sec), with some buffer time for review.

Q7: Is it necessary to write zero coefficients?

Answer: While not always mandatory, it’s highly recommended! Writing x² + 0x – 2 instead of x² – 2 helps maintain column alignment and prevents subtraction errors. It’s especially helpful for complex divisions.

Q8: What if I get a different answer?

Answer: If your answer differs, check these common error points: (1) Sign errors during subtraction, (2) Incorrect multiplication, (3) Missing terms, (4) Calculation mistakes. Redo the division carefully and verify using the Division Algorithm.

Q9: Can I use synthetic division for this question?

Answer: No, synthetic division only works when the divisor is of the form (x – a), i.e., a linear polynomial. Since our divisor is x² – 2 (degree 2), we must use polynomial long division.

Q10: What if the remainder is zero?

Answer: If remainder is zero, it means g(x) is a factor of p(x), and p(x) = g(x) × q(x). This is a special case. In our question, remainder is 7x – 9 (non-zero), so g(x) is not a factor of p(x).

Q11: How do I know when to stop dividing?

Answer: Stop when the degree of the remainder becomes less than the degree of the divisor. In this case, when remainder has degree < 2. Our remainder 7x - 9 has degree 1, which is less than 2, so we stop.

Q12: Will I lose marks if I don’t verify?

Answer: Yes, typically 0.5 marks are allocated for verification in a 3-mark question. Even if your answer is correct, not showing verification can cost you marks. Always verify using p(x) = g(x) × q(x) + r(x).

📚 Same Exercise (2.2):

• Question 1 → Divide x² + 3x + 1 by x + 2
• Question 2 → Divide x³ – 6x² + 11x – 6 by x² – 5x + 6
• Question 3 → Divide 2x⁴ – 3x³ – 3x² + 6x – 2 by x² – 2
• Question 4 → Similar polynomial division problem

📖 Other Exercises in Chapter 2:

• Exercise 2.1 → Graphical representation of polynomials
• Exercise 2.3 → Relationship between zeros and coefficients
• Exercise 2.4 → Word problems on polynomials

🎯 Similar Concept Questions:

• Chapter 2 Overview → All questions on Polynomials
• Chapter 3 → Linear equations (related algebraic concepts)
• Chapter 4 → Quadratic equations