Polynomial Addition

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Polynomial Addition · Grade 9 · Worksheet 2

1. Isabella is designing a rectangular mural for her school's art competition. The length of the mural is represented by the polynomial (5x² - 8x + 12) feet, and the width is represented by (3x² + 7x - 9) feet. The school's art director decides to add a decorative trim along the entire perimeter of the mural. What simplified polynomial expression represents the total length of trim needed in feet? Answer: 16x² - 2x + 6 Solution: Recall the formula for the perimeter of a rectangle: P = 2(length) + 2(width). Substitute the given expressions: P = 2(5x² - 8x + 12) + 2(3x² + 7x - 9). Distribute the 2: P = 10x² - 16x + 24 + 6x² + 14x - 18.
   Full step-by-step solution

   Step 1: Recall the formula for the perimeter of a rectangle: P = 2(length) + 2(width). Step 2: Substitute the given expressions: P = 2(5x² - 8x + 12) + 2(3x² + 7x - 9). Step 3: Distribute the 2: P = 10x² - 16x + 24 + 6x² + 14x - 18. Step 4: Combine like terms: x² terms: 10x² + 6x² = 16x² x terms: -16x + 14x = -2x constant terms: 24 - 18 = 6 Step 5: The simplified polynomial is: 16x² - 2x + 6. The total length of trim needed is represented by 16x² - 2x + 6 feet.

2. A tech startup is designing a new smartphone case. The case's length is represented by the polynomial (4x² - 3x + 7) cm and its width is (2x² + 5x - 2) cm. The company wants to add a protective rubber bumper that runs along the entire perimeter. What simplified polynomial expression represents the total length of bumper needed? Answer: 12x² + 4x + 10 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 - 2) Distribute the 2: P = 8x² - 6x + 14 + 4x² + 10x - 4 Combine like terms: (8x² + 4x²) + (-6x + 10x) + (14 - 4) Simplify: 12x² + 4x + 10 The answer 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 - 2) Step 3: Distribute the 2: P = 8x² - 6x + 14 + 4x² + 10x - 4 Step 4: Combine like terms: (8x² + 4x²) + (-6x + 10x) + (14 - 4) Step 5: Simplify: 12x² + 4x + 10 The answer is 12x² + 4x + 10.

3. (2x³ - 5x² + 3x - 7) + (4x³ + 2x² - 8x + 5) - (3x³ - x² + 6x - 2) = ? Answer: 3x³ - 2x² - 11x Solution: Write the expression: (2x³ - 5x² + 3x - 7) + (4x³ + 2x² - 8x + 5) - (3x³ - x² + 6x - 2) Distribute the negative sign to the third polynomial: (2x³ - 5x² + 3x - 7) + (4x³ + 2x² - 8x + 5) + (-3x³ + x² - 6x + 2) Combine x³ terms: 2x³ + 4x³ - 3x³ = 3x³ Combine x² terms: -5x² + 2x² + x² = -2x²…
   Full step-by-step solution

   Step 1: Write the expression: (2x³ - 5x² + 3x - 7) + (4x³ + 2x² - 8x + 5) - (3x³ - x² + 6x - 2) Step 2: Distribute the negative sign to the third polynomial: (2x³ - 5x² + 3x - 7) + (4x³ + 2x² - 8x + 5) + (-3x³ + x² - 6x + 2) Step 3: Combine x³ terms: 2x³ + 4x³ - 3x³ = 3x³ Step 4: Combine x² terms: -5x² + 2x² + x² = -2x² Step 5: Combine x terms: 3x - 8x - 6x = -11x Step 6: Combine constant terms: -7 + 5 + 2 = 0 Step 7: The simplified expression is: 3x³ - 2x² - 11x

4. (4x³ - 2x² + 7x - 5) + (3x³ + 5x² - 4x + 8) - (2x³ - 3x² + x - 6) = ? Answer: 5x³ + 6x² + 2x + 9 Solution: Write the expression: (4x³ - 2x² + 7x - 5) + (3x³ + 5x² - 4x + 8) - (2x³ - 3x² + x - 6) Distribute the negative sign to the third polynomial: (4x³ - 2x² + 7x - 5) + (3x³ + 5x² - 4x + 8) - 2x³ + 3x² - x + 6 Combine x³ terms: 4x³ + 3x³ - 2x³ = 5x³ Combine x² terms: -2x² + 5x² + 3x² = 6x² Combine x…
   Full step-by-step solution

   Step 1: Write the expression: (4x³ - 2x² + 7x - 5) + (3x³ + 5x² - 4x + 8) - (2x³ - 3x² + x - 6) Step 2: Distribute the negative sign to the third polynomial: (4x³ - 2x² + 7x - 5) + (3x³ + 5x² - 4x + 8) - 2x³ + 3x² - x + 6 Step 3: Combine x³ terms: 4x³ + 3x³ - 2x³ = 5x³ Step 4: Combine x² terms: -2x² + 5x² + 3x² = 6x² Step 5: Combine x terms: 7x - 4x - x = 2x Step 6: Combine constant terms: -5 + 8 + 6 = 9 Step 7: Write the final polynomial: 5x³ + 6x² + 2x + 9

5. A rectangular garden has a length of (3x² + 2x - 5) meters and a width of (2x² - x + 3) meters. The gardener wants to install a decorative border around the entire perimeter. Write a simplified polynomial expression that represents the total perimeter of the garden in meters. Answer: 10x² + 2x - 4 Solution: Recall the formula for the perimeter of a rectangle. P = 2 × (length + width) Write down the given length and width. Length = 3x² + 2x - 5 Width = 2x² - x + 3 Add the length and width.
   Full step-by-step solution

   Step 1: Recall the formula for the perimeter of a rectangle. The perimeter P of a rectangle is given by: P = 2 × (length + width) Step 2: Write down the given length and width. Length = 3x² + 2x - 5 Width = 2x² - x + 3 Step 3: Add the length and width. (3x² + 2x - 5) + (2x² - x + 3) Group like terms: 3x² + 2x² = 5x² 2x + (-x) = 2x - x = x -5 + 3 = -2 So, length + width = 5x² + x - 2 Step 4: Multiply the sum by 2 to get the perimeter. P = 2 × (5x² + x - 2) Distribute the 2: 2 × 5x² = 10x² 2 × x = 2x 2 × (-2) = -4 So, P = 10x² + 2x - 4 Step 5: Final simplified polynomial expression for the perimeter. Perimeter = 10x² + 2x - 4 meters

6. A tech startup 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 perimeter for the screen bezel. What simplified polynomial expression represents the perimeter of the smartphone screen in centimeters? Answer: 12x² + 4x + 6 Solution: Recall the perimeter formula for a rectangle: P = 2 × length + 2 × width Substitute the given expressions: 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 perimeter formula for a rectangle: P = 2 × length + 2 × width Step 2: Substitute the given expressions: 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

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

   Step 1: Write the expression: (4x³ - 2x² + 7x - 5) + (3x³ + 5x² - 4x + 8) - (2x³ - 3x² + 6x - 1) Step 2: Distribute the negative sign: 4x³ - 2x² + 7x - 5 + 3x³ + 5x² - 4x + 8 - 2x³ + 3x² - 6x + 1 Step 3: Combine x³ terms: 4x³ + 3x³ - 2x³ = 5x³ Step 4: Combine x² terms: -2x² + 5x² + 3x² = 6x² Step 5: Combine x terms: 7x - 4x - 6x = -3x Step 6: Combine constant terms: -5 + 8 + 1 = 4 Step 7: Final answer: 5x³ + 6x² - 3x + 4

8. (6x³ - 11x² + 16x - 1) + (x³ + 6x² - 11x + 6) - (2x³ - 6x² + x - 1) = ? Answer: 5x³ + x² + 4x + 6 Solution: Write the expression: (6x³ - 11x² + 16x - 1) + (x³ + 6x² - 11x + 6) - (2x³ - 6x² + x - 1) Distribute the negative sign to the third polynomial: (6x³ - 11x² + 16x - 1) + (x³ + 6x² - 11x + 6) + (-2x³ + 6x² - x + 1) Combine x³ terms: 6x³ + x³ - 2x³ = 5x³ Combine x² terms: -11x² + 6x² + 6x² = 1x²…
   Full step-by-step solution

   Step 1: Write the expression: (6x³ - 11x² + 16x - 1) + (x³ + 6x² - 11x + 6) - (2x³ - 6x² + x - 1) Step 2: Distribute the negative sign to the third polynomial: (6x³ - 11x² + 16x - 1) + (x³ + 6x² - 11x + 6) + (-2x³ + 6x² - x + 1) Step 3: Combine x³ terms: 6x³ + x³ - 2x³ = 5x³ Step 4: Combine x² terms: -11x² + 6x² + 6x² = 1x² (or just x²) Step 5: Combine x terms: 16x - 11x - x = 4x Step 6: Combine constant terms: -1 + 6 + 1 = 6 Step 7: The simplified expression is: 5x³ + x² + 4x + 6