Case Study Polynomials class 9

Case Study Polynomials class 9

Case Study Mathematics Class 9 Polynomial | Free Online Test

Case Study Polynomials Class 9: Introduction

Students preparing for exams often search for Case Study math questions for class 9. These questions improve analytical skills and help understand complex concepts. Our online test platform provides interactive exercises on polynomials specifically designed for Class 9 students. Practicing these problems regularly boosts confidence and speeds up problem-solving. Additionally, these exercises make learning engaging and help in applying formulas effectively.

Importance of Math Case Study Questions Class 9

Math case study questions class 9 focus on real-world scenarios where polynomials are used. They encourage critical thinking. For example, finding roots or factoring equations in context enhances understanding. Moreover, students can learn step-by-step problem-solving techniques. Transition words like “furthermore” and “therefore” guide learners through logical steps easily.

Online Test Benefits

Our math case study questions online test allows instant feedback. Students can identify mistakes quickly. The tests include Case Study Polynomials class 9 exercises covering various difficulty levels. Short questions help reinforce key concepts. Consequently, consistent practice ensures better exam performance and conceptual clarity.

Case Study: Remainder Theorem and Zeroes of Polynomials

Case Study 2: Remainder Theorem and Zeroes of Polynomials in Real-Life Context

A student working on a science project wants to analyze a mathematical function related to the speed of a rotating disc. She models the velocity of the disc using the polynomial:

\[ f(x) = x^3 – 6x^2 + 11x – 6 \]
To study how efficiently the disc slows down, she looks for the zeroes of the polynomial where the velocity becomes zero. Her teacher introduces her to the Remainder Theorem, which helps find whether a value is a zero of a polynomial.

Remainder Theorem: If \( f(x) \) is divided by \( (x – a) \), then the remainder is \( f(a) \). If \( f(a) = 0 \), then \( (x – a) \) is a factor of \( f(x) \).

Using this theorem and the Factor Theorem, she begins analyzing how the roots of polynomials can be verified and used in physical modeling.

Useful Concepts and Equations:

  • Zero of a polynomial: A value of \( x \) for which \( f(x) = 0 \).
  • Factor Theorem: \( (x – a) \) is a factor of \( f(x) \) if and only if \( f(a) = 0 \).
  • Division Algorithm: If \( f(x) \) and \( g(x) \) are polynomials, then \( f(x) = g(x)q(x) + r(x) \) where \( r(x) = 0 \) or \( \deg(r) < \deg(g) \).

1. Using the Remainder Theorem, what is the remainder when \( f(x) = x^3 – 6x^2 + 11x – 6 \) is divided by \( (x – 1) \)?

  • A) 0
  • B) 1
  • C) -1
  • D) 5
Answer: A) 0
Solution: \( f(1) = 1^3 – 6(1)^2 + 11(1) – 6 = 1 – 6 + 11 – 6 = 0 \).

2. Which of the following is a factor of \( f(x) = x^3 – 6x^2 + 11x – 6 \)?

  • A) \( (x – 4) \)
  • B) \( (x – 1) \)
  • C) \( (x + 1) \)
  • D) \( (x – 5) \)
Answer: B) \( (x – 1) \)
Solution: As shown earlier, \( f(1) = 0 \) so \( (x – 1) \) is a factor.

3. Which of the following are all zeroes of \( f(x) = x^3 – 6x^2 + 11x – 6 \)?

  • A) 1, 2, 3
  • B) 2, 3, 4
  • C) 1, 2, 4
  • D) 1, 3, 5
Answer: A) 1, 2, 3
Solution: Factor \( f(x) = (x – 1)(x – 2)(x – 3) \Rightarrow \) zeroes are 1, 2, 3.

4. What is the value of \( f(2) \) if \( f(x) = x^3 – 6x^2 + 11x – 6 \)?

  • A) 1
  • B) -1
  • C) 0
  • D) 2
Answer: C) 0
Solution: \( f(2) = 8 – 24 + 22 – 6 = 0 \)

5. If \( (x – 2) \) is a factor of \( f(x) \), what must be true?

  • A) \( f(2) = 2 \)
  • B) \( f(2) = 1 \)
  • C) \( f(2) = 0 \)
  • D) \( f(2) = -1 \)
Answer: C) \( f(2) = 0 \)
Solution: By Factor Theorem, if \( (x – 2) \) is a factor, then \( f(2) = 0 \).

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