Polynomial Addition

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Polynomial Addition · Grade 9 · Worksheet 3

1. Mason is designing a geometric mosaic on a coordinate grid. The perimeter of the mosaic is defined by four vertices: A (0, 0), B (2x² + 7, 0), C (2x² + 7, 2x - 7), and D (0, 2x - 7). A smaller triangular section is removed from one corner, with vertices at (0, 0), (x² + 2, 0), and (0, x - 2). What polynomial expression represents the remaining area of the mosaic after the triangular section is removed? Answer: 2x³ + 14x² - 7x - 49 - (1/2)x³ - x² - 2x + 2 simplifies to (3/2)x³ + 13x² - 9x - 47 Solution: Find the area of the rectangle. The rectangle has length from x=0 to x=2x²+7, so length = 2x²+7. The height from y=0 to y=2x-7, so height = 2x-7.
   Full step-by-step solution

   Step 1: Find the area of the rectangle. The rectangle has length from x=0 to x=2x²+7, so length = 2x²+7. The height from y=0 to y=2x-7, so height = 2x-7. Area of rectangle = length * height = (2x²+7)(2x-7). Multiply: 2x²(2x-7) = 4x³ - 14x². Then 7(2x-7) = 14x - 49. Total rectangle area = 4x³ - 14x² + 14x - 49. Step 2: Find the area of the triangle. The triangle vertices are (0,0), (x²+2,0), and (0,x-2). Base along x-axis from 0 to x²+2, so base = x²+2. Height along y-axis from 0 to x-2, so height = x-2. Area of triangle = (1/2) * base * height = (1/2)(x²+2)(x-2). Multiply (x²+2)(x-2) = x²(x-2) + 2(x-2) = x³ - 2x² + 2x - 4. Then half gives (1/2)x³ - x² + x - 2. Step 3: Subtract triangle area from rectangle area. Remaining area = (4x³ - 14x² + 14x - 49) - ((1/2)x³ - x² + x - 2). Distribute the minus: 4x³ - 14x² + 14x - 49 - (1/2)x³ + x² - x + 2. Combine like terms: x³ terms: 4x³ - (1/2)x³ = (8/2 - 1/2)x³ = (7/2)x³. x² terms: -14x² + x² = -13x². x terms: 14x - x = 13x. Constants: -49 + 2 = -47. So remaining area = (7/2)x³ - 13x² + 13x - 47. The answer is (7/2)x³ - 13x² + 13x - 47.

2. A tech startup is modeling their quarterly revenue using polynomial functions. Their revenue from software sales is represented by R_s(x) = 4x³ - 2x² + 5x - 3, while their hardware revenue is R_h(x) = -2x³ + 7x² - 4x + 8, where x represents the quarter number. What polynomial expression represents their total quarterly revenue? Answer: 2x^3 + 5x^2 + x + 5 Solution: When adding polynomials representing different quantities, combine like terms by adding their coefficients. This is similar to how in business, total revenue equals the sum of revenues from all product lines.
   Full step-by-step solution

   When adding polynomials representing different quantities, combine like terms by adding their coefficients. This is similar to how in business, total revenue equals the sum of revenues from all product lines. The process involves grouping terms with the same variable and exponent, then performing the addition operation on their coefficients.

3. A technology company is designing a new smartphone with a rectangular screen. The screen's length is represented by the polynomial (4x² - 3x + 7) millimeters and its width is (2x² + 5x - 4) millimeters. The engineers need to calculate the total length of the protective bezel that will frame the screen. What polynomial expression represents the perimeter of the smartphone screen in millimeters? Answer: 12x² + 4x + 6 Solution: The perimeter of a rectangle is calculated using the formula P = 2l + 2w, where l represents the length and w represents the width. This formula works because a rectangle has two equal lengths and two equal widths.
   Full step-by-step solution

   The perimeter of a rectangle is calculated using the formula P = 2l + 2w, where l represents the length and w represents the width. This formula works because a rectangle has two equal lengths and two equal widths. When working with polynomials, you distribute the multiplication across all terms before combining like terms.

4. Matiu is designing a rectangular mosaic on a coordinate grid. The top-left corner of the mosaic is at (0, 0) and the bottom-right corner is at (4x² + 6x - 8, 2x² - 4x + 10). A triangular tile pattern is placed inside the mosaic with vertices at (0, 0), (4x² + 6x - 8, 0), and (0, 2x² - 4x + 10). What polynomial expression represents the perimeter of the entire rectangular mosaic? Answer: 12x² + 4x + 4 Solution: Identify the length and width from the coordinates. The length (horizontal side) is the x-coordinate of the bottom-right corner: L = 4x² + 6x - 8.
   Full step-by-step solution

   Step 1: Identify the length and width from the coordinates. The length (horizontal side) is the x-coordinate of the bottom-right corner: L = 4x² + 6x - 8. The width (vertical side) is the y-coordinate of the bottom-right corner: W = 2x² - 4x + 10. Step 2: Write the formula for the perimeter of a rectangle. Perimeter P = 2(L + W). Step 3: Add the length and width polynomials. L + W = (4x² + 6x - 8) + (2x² - 4x + 10) Combine like terms: - x² terms: 4x² + 2x² = 6x² - x terms: 6x + (-4x) = 2x - Constant terms: -8 + 10 = 2 So L + W = 6x² + 2x + 2. Step 4: Multiply the sum by 2 to find the perimeter. P = 2(6x² + 2x + 2) P = 12x² + 4x + 4. The perimeter of the rectangular mosaic is 12x² + 4x + 4.

5. A tech company is designing a new smartphone with a rectangular screen. The screen's length is represented by the polynomial (4x² - 3x + 7) centimeters and its width is (2x² + 5x - 4) centimeters. The engineers need to calculate the total length of the protective bezel that will surround the screen. What polynomial expression represents the perimeter of the smartphone screen in centimeters? Answer: 12x² + 4x + 6 Solution: Recall the formula for perimeter of a rectangle: P = 2(length) + 2(width) Substitute the given polynomials: P = 2(4x² - 3x + 7) + 2(2x² + 5x - 4) Distribute the 2: P = 8x² - 6x + 14 + 4x² + 10x - 8 Combine like terms: (8x² + 4x²) + (-6x + 10x) + (14 - 8) Simplify: 12x² + 4x + 6 The perimeter is…
   Full step-by-step solution

   Step 1: Recall the formula for perimeter of a rectangle: P = 2(length) + 2(width) Step 2: Substitute the given polynomials: P = 2(4x² - 3x + 7) + 2(2x² + 5x - 4) Step 3: Distribute the 2: P = 8x² - 6x + 14 + 4x² + 10x - 8 Step 4: Combine like terms: (8x² + 4x²) + (-6x + 10x) + (14 - 8) Step 5: Simplify: 12x² + 4x + 6 Step 6: The perimeter is 12x² + 4x + 6 centimeters.

6. Sophia is a structural engineer designing a steel support beam for a bridge. The total length of the beam is represented by the polynomial (6x² + 11x - 6) meters. Due to a design revision, she must cut off a section measuring (x² - 6x + 1) meters from one end and then weld an additional section measuring (x² + 6x - 1) meters to the other end. What simplified polynomial expression represents the final length of the beam in meters? Answer: 6x² + 23x - 6 Solution: Start with the original length: (6x² + 11x - 6) Subtract the cut section: (6x² + 11x - 6) - (x² - 6x + 1) Distribute the subtraction: 6x² + 11x - 6 - x² + 6x - 1 Combine like terms: (6x² - x²) + (11x + 6x) + (-6 - 1) = 5x² + 17x - 7 Add the welded section: (5x² + 17x - 7) + (x² + 6x - 1) Combine…
   Full step-by-step solution

   Step 1: Start with the original length: (6x² + 11x - 6) Step 2: Subtract the cut section: (6x² + 11x - 6) - (x² - 6x + 1) Distribute the subtraction: 6x² + 11x - 6 - x² + 6x - 1 Combine like terms: (6x² - x²) + (11x + 6x) + (-6 - 1) = 5x² + 17x - 7 Step 3: Add the welded section: (5x² + 17x - 7) + (x² + 6x - 1) Combine like terms: (5x² + x²) + (17x + 6x) + (-7 - 1) = 6x² + 23x - 8 Step 4: The final simplified polynomial is 6x² + 23x - 8. The answer is 6x² + 23x - 8.