Case Study Mathematics Class 9 Polynomial

Case Study 1: Understanding Polynomial Types and Operations

An engineer is designing a parabolic bridge arch. To model the height of the bridge at any horizontal distance \( x \) from one end, she uses a quadratic polynomial function: \[ h(x) = -2x^2 + 8x + 3 \] This polynomial gives the height in meters. She also compares other polynomial models used in earlier bridge designs, such as \( f(x) = 3x – 5 \) (linear), \( g(x) = x^3 – 2x^2 + x – 4 \) (cubic), and \( k(x) = 7 \) (constant). To analyze the properties of these polynomial functions, she looks at their degrees, coefficients, terms, and possible operations (addition, subtraction, multiplication). She also needs to factor expressions using standard identities and check the number of zeros.

Some useful concepts and formulas:

• A polynomial is an expression of the form: \( a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0 \) where \( a_i \) are real coefficients.
• Degree: The highest power of the variable \( x \) in the polynomial.
• Identities used for factorization:
  • \( (x + y)^2 = x^2 + 2xy + y^2 \)
  • \( (x – y)^2 = x^2 – 2xy + y^2 \)
  • \( x^2 – y^2 = (x + y)(x – y) \)
  • \( (x + a)(x + b) = x^2 + (a + b)x + ab \)