Chapter 2 Polynomials

Case Study 1: Polynomial Expressions in Area and Volume Calculations

Case Description:
A construction company is designing a rectangular garden with a length represented by the polynomial $l(x) = x + 3$ meters and a width represented by the polynomial $w(x) = x - 2$ meters. The company is also creating a walkway around the garden. The area of the garden can be represented by the product of these two polynomials, which allows the company to calculate the area for different values of $x$. To ensure enough space, they also want to calculate the perimeter based on these polynomial expressions.

MCQs:

1. What is the polynomial expression for the area of the garden, $A(x)$?

   • A) $x^2 + x - 6$
   • B) $x^2 + 6x - 6$
   • C) $x^2 + x + 6$
   • D) $x^2 + 5x - 6$

2. What is the perimeter of the garden, expressed as a polynomial $P(x)$?

   • A) $4x + 6$
   • B) $2x + 6$
   • C) $2x + 2$
   • D) $2x + 2x - 4$

3. If $x = 5$, what would be the area of the garden?

   • A) 35 square meters
   • B) 45 square meters
   • C) 55 square meters
   • D) 65 square meters

4. For $x = 2$, what is the perimeter of the garden?

   • A) 14 meters
   • B) 18 meters
   • C) 20 meters
   • D) 24 meters

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Case Study 2: Nature of Roots of a Quadratic Polynomial

Case Description:
In a science experiment, a ball is thrown vertically into the air from a platform 5 meters high, and its height at any time $t$ seconds is given by the polynomial $h(t) = -5t^2 + 20t + 5$. The experiment aims to determine the time when the ball will reach the maximum height, return to the ground, and find the nature of the polynomial’s roots. This experiment helps students connect the concept of quadratic polynomials to real-life scenarios and understand the nature of roots through graphical analysis.

MCQs:

1. What is the degree of the polynomial $h(t) = -5t^2 + 20t + 5$?

   • A) 1
   • B) 2
   • C) 3
   • D) 0

2. What type of roots does this polynomial have?

   • A) Real and distinct
   • B) Real and equal
   • C) Imaginary
   • D) None

3. What is the maximum height reached by the ball?

   • A) 10 meters
   • B) 15 meters
   • C) 20 meters
   • D) 25 meters

4. When the ball hits the ground, which value of $t$ solves $h(t) = 0$?

   • A) 2 seconds
   • B) 4 seconds
   • C) 6 seconds
   • D) 8 seconds

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Case Study 3: Using Factor Theorem in Real-Life Situations

Case Description:
A farmer is designing a rectangular field whose area in square meters is given by the polynomial $A(x) = x^2 - 9x + 20$. The farmer decides to find the dimensions of the field such that the factors of this polynomial can represent the length and width of the field. By applying the Factor Theorem, the farmer identifies the dimensions and checks if the polynomial can be factored into linear expressions. This approach helps the farmer practically understand the Factor Theorem and solve real-life problems.

MCQs:

1. What are the factors of the polynomial $A(x) = x^2 - 9x + 20$?

   • A) $(x - 4)(x - 5)$
   • B) $(x + 4)(x + 5)$
   • C) $(x - 2)(x - 10)$
   • D) $(x + 2)(x + 10)$

2. What are the possible dimensions of the field if $x$ is a positive integer?

   • A) 5 m by 6 m
   • B) 4 m by 5 m
   • C) 3 m by 8 m
   • D) 7 m by 3 m

3. If the polynomial has a root $x = 5$, what other root will satisfy the equation $A(x) = 0$?

   • A) 2
   • B) 4
   • C) 5
   • D) 6

4. What is the length of the field when $x = 10$?

   • A) 20 meters
   • B) 35 meters
   • C) 50 meters
   • D) 80 meters

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Case Study 4: Relationship between Coefficients and Roots of a Polynomial

Case Description:
In a chemistry experiment, the rate of reaction is expressed by a quadratic polynomial $r(x) = 2x^2 - 5x + 3$, where $x$ represents the concentration of a certain substance. By adjusting the concentration, the experiment aims to determine the time when the reaction rate is maximized. The relationship between the polynomial’s coefficients and its roots helps the students understand how changes in concentration affect the reaction rate.

MCQs:

1. What is the sum of the roots of the polynomial $r(x) = 2x^2 - 5x + 3$?

   • A) $\frac{5}{2}$
   • B) -5
   • C) 5
   • D) -2.5

2. What is the product of the roots of $r(x) = 2x^2 - 5x + 3$?

   • A) 3
   • B) -3
   • C) 1.5
   • D) $\frac{3}{2}$

3. Which equation represents a scenario where the roots are equal?

   • A) $r(x) = x^2 - 4x + 4$
   • B) $r(x) = 3x^2 + 2x + 3$
   • C) $r(x) = x^2 + 2x + 3$
   • D) $r(x) = 2x^2 - 8x + 4$

4. What would be the roots of the polynomial if $r(x) = x^2 - 4x + 3$?

   • A) $(1, 3)$
   • B) $(2, 2)$
   • C) $(1, -3)$
   • D) $(2, 3)$

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Case Study 5: Using Polynomial Functions in Business Profit Analysis

Case Description:
A bakery models its monthly profit $P(x)$, based on the number of cakes sold, with the polynomial $P(x) = -3x^2 + 30x - 72$. The bakery owners want to determine how many cakes they need to sell to maximize profit. By using the concepts of polynomials, they aim to identify the maximum point of the profit function, analyze profit at specific sales volumes, and make informed decisions on pricing and production to increase their profits.

MCQs:

1. What type of polynomial is $P(x) = -3x^2 + 30x - 72$?

   • A) Linear
   • B) Quadratic
   • C) Cubic
   • D) Constant

2. To find the maximum profit, the bakery needs to:

   • A) Find the roots of the polynomial
   • B) Find the vertex of the parabola represented by the polynomial
   • C) Find the degree of the polynomial
   • D) Solve $P(x) = 0$

3. At what value of $x$ does the bakery reach its maximum profit?

   • A) $x = 5$
   • B) $x = 6$
   • C) $x = 10$
   • D) $x = 15$

4. What will be the maximum profit for the bakery?

   • A) 150
   • B) 225
   • C) 250
   • D) 200