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Real-World Polynomials: Practical Applications and Problem-Solving Techniques

Explore real-world scenarios involving polynomials, such as calculating perimeters, volumes, profits, and areas, to enhance your understanding of polynomial functions.

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Real-World Polynomials: Practical Applications and Problem-Solving Techniques

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  1. Real-World Polynomials Each side of an equilateral triangle has length w + 2. Each side of a square has length 3w – 4. Write an expression for the sum of the perimeter of the equilateral triangle and the perimeter of the square. Perimeter of triangle = w + 2 + w + 2 + w + 2

  2. Real-World Polynomials Each side of an equilateral triangle has length w + 2. Each side of a square has length 3w – 4. Write an expression for the sum of the perimeter of the equilateral triangle and the perimeter of the square. Perimeter of square = 3w – 4 + 3w – 4 + 3w – 4 + 3w – 4

  3. Real-World Polynomials Each side of an equilateral triangle has length w + 2. Each side of a square has length 3w – 4. Write an expression for the sum of the perimeter of the equilateral triangle and the perimeter of the square. Sum of perimeters = Perimeter of Tri. + Square = 3w + 6 + 12w – 16 = 15w – 10

  4. Real-World Polynomials The volume of a rectangular prism is given by the expression x3 + 2x2 – 4x + 6. The volume of a smaller rectangular prism is given by the expression 4x3 – 5x2 + 6x – 12. How much greater is the volume of the larger prism? x3 + 2x2 – 4x + 6 – (4x3 – 5x2 + 6x – 12) = -3x3 + 7x2 – 10x + 18

  5. Real-World Polynomials Suppose the cost in dollars of producing x model kits is given by the polynomial 500,000 + 2x and the revenue generated from sales is given by the polynomial 30x – 0.00005x2. Find a polynomial expression for the profit from making and selling x model kits, and evaluate the expression for x = 300,000. Profit = Revenue – Cost = 30x – 0.00005x2 – (500,000 + 2x) = -0.00005x2 + 28x – 500,000 When x = 300,000 Profit = 3,400,000

  6. Real-World Polynomials Libby is putting a mat of width m and a frame of width f around an 8-inch by 10-inch picture. Write an expression for the perimeter of the frame.

  7. Real-World Polynomials Libby is putting a mat of width m and a frame of width f around an 8-inch by 10-inch picture. Write an expression for the perimeter of the frame. 8f + 8m + 36

  8. Real-World Polynomials Colette is putting a mat of width 3w and a frame of width w around a 16-inch by 48-inch poster. Find an expression for the perimeter of frame. Length 1 = 8w + 16 Length 2 = 8w + 48 Perimeter = 32w + 128

  9. Real-World Polynomials A rectangle has length x + 5 and width x – 3. What is the area of the rectangle? Area = (x + 5)(x – 3) = x2 + 2x – 15

  10. Real-World Polynomials The figures below are squares. Find an expression for the area of the shaded region. Area = (x + 4)(x + 4) – (x – 1)(x – 1) = x2 + 16x + 16 – (x2 – 2x + 1) = 18x + 15

  11. Real-World Polynomials The figures below are squares. Find an expression for the area of the shaded region. Area = (x + 3)(x + 3) – (x)(x) = x2 + 6x + 9 – (x2) = 6x + 9

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