Polynomial Addition

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Polynomial Addition · Grade 9 · Worksheet 3

1. A triangular garden plot is drawn on a coordinate grid. The vertices are at A (2x - 3, 0), B (0, 4x + 7), and C (x^2 + 2x, x^2 - 3x + 2). A walkway runs along the perimeter of the triangle. The length of side AB is (5x^2 - 7x + 2) units, side BC is (3x^2 + 4x - 7) units, and side CA is (2x^2 + 7x + 2) units. What is the total length of the walkway around the triangular garden plot, expressed as a simplified polynomial? Answer: 10x^2 + 4x - 3 Solution: Write the three side lengths. Side AB = 5x^2 - 7x + 2 Side BC = 3x^2 + 4x - 7 Side CA = 2x^2 + 7x + 2 Add the polynomials by combining like terms.
   Full step-by-step solution

   Step 1: Write the three side lengths. Side AB = 5x^2 - 7x + 2 Side BC = 3x^2 + 4x - 7 Side CA = 2x^2 + 7x + 2 Step 2: Add the polynomials by combining like terms. Total = (5x^2 - 7x + 2) + (3x^2 + 4x - 7) + (2x^2 + 7x + 2) Step 3: Group the x^2 terms: 5x^2 + 3x^2 + 2x^2 = 10x^2 Step 4: Group the x terms: -7x + 4x + 7x = 4x Step 5: Group the constant terms: 2 - 7 + 2 = -3 Step 6: Combine: Total = 10x^2 + 4x - 3 The answer is 10x^2 + 4x - 3.

2. Aroha is a landscape architect designing a park that features two adjacent rectangular flower beds. The length of the first flower bed is represented by the polynomial (9x² - 11x + 13) meters, and its width is (7x² + 10x - 8) meters. The second flower bed has a length of (6x² + 14x - 9) meters and a width of (5x² - 12x + 15) meters. The city council wants to install an LED light strip around the combined perimeter of both flower beds. What simplified polynomial expression represents the total length of light strip needed, in meters? Answer: 54x² + 2x + 22 Solution: Recall the perimeter formula for a rectangle: P = 2(length) + 2(width) For the first flower bed, length = 9x² - 11x + 13, width = 7x² + 10x - 8.
   Full step-by-step solution

   Step 1: Recall the perimeter formula for a rectangle: P = 2(length) + 2(width) Step 2: For the first flower bed, length = 9x² - 11x + 13, width = 7x² + 10x - 8. So P₁ = 2(9x² - 11x + 13) + 2(7x² + 10x - 8) Step 3: Distribute the 2 in P₁: 18x² - 22x + 26 + 14x² + 20x - 16 Step 4: Combine like terms in P₁: (18x² + 14x²) + (-22x + 20x) + (26 - 16) = 32x² - 2x + 10 Step 5: For the second flower bed, length = 6x² + 14x - 9, width = 5x² - 12x + 15. So P₂ = 2(6x² + 14x - 9) + 2(5x² - 12x + 15) Step 6: Distribute the 2 in P₂: 12x² + 28x - 18 + 10x² - 24x + 30 Step 7: Combine like terms in P₂: (12x² + 10x²) + (28x - 24x) + (-18 + 30) = 22x² + 4x + 12 Step 8: Add the perimeters: Total = P₁ + P₂ = (32x² - 2x + 10) + (22x² + 4x + 12) Step 9: Combine like terms: (32x² + 22x²) + (-2x + 4x) + (10 + 12) = 54x² + 2x + 22 The answer is 54x² + 2x + 22.

3. A rectangular mosaic panel designed by Mere has its length represented by the polynomial (4x² + 6x - 8) centimeters and its width represented by (2x² - 4x + 10) centimeters. A square decorative tile is removed from the center of the panel. The side length of the square tile is represented by the polynomial (x² - 2x + 4) centimeters. What polynomial expression represents the remaining area of the mosaic panel after the square tile is removed? Answer: 7x² + 12x - 24 Solution: Find the area of the rectangular panel. Area of rectangle = length * width = (4x² + 6x - 8) * (2x² - 4x + 10) 4x² * 2x² = 8x⁴ 4x² * (-4x) = -16x³ 4x² * 10 = 40x² 6x * 2x² = 12x³ 6x * (-4x) = -24x² 6x * 10 = 60x -8 * 2x² = -16x² -8 * (-4x) = 32x -8 * 10 = -80 x⁴: 8x⁴ x³: -16x³ + 12x³ = -4x³ x²:…
   Full step-by-step solution

   Step 1: Find the area of the rectangular panel. Area of rectangle = length * width = (4x² + 6x - 8) * (2x² - 4x + 10) Multiply each term: 4x² * 2x² = 8x⁴ 4x² * (-4x) = -16x³ 4x² * 10 = 40x² 6x * 2x² = 12x³ 6x * (-4x) = -24x² 6x * 10 = 60x -8 * 2x² = -16x² -8 * (-4x) = 32x -8 * 10 = -80 Combine like terms: x⁴: 8x⁴ x³: -16x³ + 12x³ = -4x³ x²: 40x² - 24x² - 16x² = 0x² x: 60x + 32x = 92x constant: -80 So rectangle area = 8x⁴ - 4x³ + 92x - 80 Step 2: Find the area of the square tile. Area of square = side * side = (x² - 2x + 4) * (x² - 2x + 4) Multiply each term: x² * x² = x⁴ x² * (-2x) = -2x³ x² * 4 = 4x² (-2x) * x² = -2x³ (-2x) * (-2x) = 4x² (-2x) * 4 = -8x 4 * x² = 4x² 4 * (-2x) = -8x 4 * 4 = 16 Combine like terms: x⁴: x⁴ x³: -2x³ - 2x³ = -4x³ x²: 4x² + 4x² + 4x² = 12x² x: -8x - 8x = -16x constant: 16 So square area = x⁴ - 4x³ + 12x² - 16x + 16 Step 3: Subtract the square area from the rectangle area. Remaining area = (8x⁴ - 4x³ + 92x - 80) - (x⁴ - 4x³ + 12x² - 16x + 16) = 8x⁴ - 4x³ + 92x - 80 - x⁴ + 4x³ - 12x² + 16x - 16 Combine like terms: x⁴: 8x⁴ - x⁴ = 7x⁴ x³: -4x³ + 4x³ = 0x³ x²: -12x² x: 92x + 16x = 108x constant: -80 - 16 = -96 So remaining area = 7x⁴ - 12x² + 108x - 96 The answer is 7x⁴ - 12x² + 108x - 96.

4. Tane is analyzing the profit of his two businesses. The profit from his coffee shop is modeled by the polynomial (9x² + 13x - 15) dollars, where x is the number of months since opening. The profit from his bookstore is modeled by (-4x² + 11x + 20) dollars. Tane wants to find the total profit from both businesses combined. What simplified polynomial expression represents the total profit? Answer: 5x² + 24x + 5 Solution: Write the sum of the two profit polynomials: (9x² + 13x - 15) + (-4x² + 11x + 20) Group like terms: (9x² + (-4x²)) + (13x + 11x) + (-15 + 20) Combine the x² terms: 9x² - 4x² = 5x² Combine the x terms: 13x + 11x = 24x Combine the constant terms: -15 + 20 = 5 Write the simplified polynomial: 5x² +…
   Full step-by-step solution

   Step 1: Write the sum of the two profit polynomials: (9x² + 13x - 15) + (-4x² + 11x + 20) Step 2: Group like terms: (9x² + (-4x²)) + (13x + 11x) + (-15 + 20) Step 3: Combine the x² terms: 9x² - 4x² = 5x² Step 4: Combine the x terms: 13x + 11x = 24x Step 5: Combine the constant terms: -15 + 20 = 5 Step 6: Write the simplified polynomial: 5x² + 24x + 5 The total profit is 5x² + 24x + 5 dollars.

5. Emma is tracking the growth of two experimental plant species over time. The height of Plant A is modeled by the polynomial (5x² - 3x + 9) centimeters, and the height of Plant B is modeled by (3x² + 7x - 5) centimeters, where x represents the number of weeks since the experiment began. Emma wants to find the combined height of both plants after x weeks. What simplified polynomial expression represents the total height of the two plants? Answer: 8x² + 4x + 4 Solution: Write the expression for the sum of the two heights: (5x² - 3x + 9) + (3x² + 7x - 5) Remove parentheses: 5x² - 3x + 9 + 3x² + 7x - 5 Group like terms: (5x² + 3x²) + (-3x + 7x) + (9 - 5) Combine like terms: 8x² + 4x + 4 The simplified polynomial expression for the total height is 8x² + 4x + 4.
   Full step-by-step solution

   Step 1: Write the expression for the sum of the two heights: (5x² - 3x + 9) + (3x² + 7x - 5) Step 2: Remove parentheses: 5x² - 3x + 9 + 3x² + 7x - 5 Step 3: Group like terms: (5x² + 3x²) + (-3x + 7x) + (9 - 5) Step 4: Combine like terms: 8x² + 4x + 4 The simplified polynomial expression for the total height is 8x² + 4x + 4.

6. (4x³ - 2x² + 7x - 5) + (3x³ + 5x² - 2x + 8) - (2x³ - 4x² + 3x - 1) = ? Answer: 5x³ + 7x² + 2x + 4 Solution: Write the expression: (4x³ - 2x² + 7x - 5) + (3x³ + 5x² - 2x + 8) - (2x³ - 4x² + 3x - 1) Distribute the negative sign to the third polynomial: (4x³ - 2x² + 7x - 5) + (3x³ + 5x² - 2x + 8) + (-2x³ + 4x² - 3x + 1) Combine x³ terms: 4x³ + 3x³ - 2x³ = 5x³ Combine x² terms: -2x² + 5x² + 4x² = 7x²…
   Full step-by-step solution

   Step 1: Write the expression: (4x³ - 2x² + 7x - 5) + (3x³ + 5x² - 2x + 8) - (2x³ - 4x² + 3x - 1) Step 2: Distribute the negative sign to the third polynomial: (4x³ - 2x² + 7x - 5) + (3x³ + 5x² - 2x + 8) + (-2x³ + 4x² - 3x + 1) Step 3: Combine x³ terms: 4x³ + 3x³ - 2x³ = 5x³ Step 4: Combine x² terms: -2x² + 5x² + 4x² = 7x² Step 5: Combine x terms: 7x - 2x - 3x = 2x Step 6: Combine constant terms: -5 + 8 + 1 = 4 Step 7: Write the final simplified polynomial: 5x³ + 7x² + 2x + 4