Polynomial Addition

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Polynomial Addition · Grade 9 · Worksheet 1

1. (9x³ - 14x² + 22x - 8) + (12x³ + 8x² - 17x + 15) - (7x³ - 11x² + 9x - 4) = ? Answer: 14x³ + 5x² - 4x + 11 Solution: Write the expression: (9x³ - 14x² + 22x - 8) + (12x³ + 8x² - 17x + 15) - (7x³ - 11x² + 9x - 4) Distribute the subtraction sign to the third polynomial: (9x³ - 14x² + 22x - 8) + (12x³ + 8x² - 17x + 15) + (-7x³ + 11x² - 9x + 4) Combine x³ terms: 9x³ + 12x³ - 7x³ = 14x³ Combine x² terms: -14x² +…
   Full step-by-step solution

   Step 1: Write the expression: (9x³ - 14x² + 22x - 8) + (12x³ + 8x² - 17x + 15) - (7x³ - 11x² + 9x - 4) Step 2: Distribute the subtraction sign to the third polynomial: (9x³ - 14x² + 22x - 8) + (12x³ + 8x² - 17x + 15) + (-7x³ + 11x² - 9x + 4) Step 3: Combine x³ terms: 9x³ + 12x³ - 7x³ = 14x³ Step 4: Combine x² terms: -14x² + 8x² + 11x² = 5x² Step 5: Combine x terms: 22x - 17x - 9x = -4x Step 6: Combine constant terms: -8 + 15 + 4 = 11 Step 7: Write the final answer: 14x³ + 5x² - 4x + 11

2. 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.

3. 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

4. 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 - 7, where x represents the number of units sold in thousands. Their hardware revenue is given by R_h(x) = -2x³ + 3x² - x + 4. The company wants to analyze their total revenue function R_total(x) = R_s(x) + R_h(x). What polynomial represents their combined quarterly revenue? Answer: 2x³ + x² + 4x - 3 Solution: Polynomial addition involves combining like terms, which are terms with the same variable raised to the same power.
   Full step-by-step solution

   Polynomial addition involves combining like terms, which are terms with the same variable raised to the same power. This concept is frequently used in business and economics when modeling combined revenue, costs, or profits from multiple sources. The coefficients represent the contribution of each term to the total, and adding polynomials allows us to create a unified model from separate components.

5. (6x³ - 11x² + 16x - 1) + (x³ + 6x² - 11x + 6) - (2x³ - x² + 6x - 1) = ? Answer: 5x³ - 4x² - x + 6 Solution: Write the expression without parentheses, distributing the negative sign to the third polynomial: (6x³ - 11x² + 16x - 1) + (x³ + 6x² - 11x + 6) + (-2x³ + x² - 6x + 1) Combine x³ terms: 6x³ + x³ - 2x³ = 5x³ Combine x² terms: -11x² + 6x² + x² = -4x² Combine x terms: 16x - 11x - 6x = -x Combine…
   Full step-by-step solution

   Step 1: Write the expression without parentheses, distributing the negative sign to the third polynomial: (6x³ - 11x² + 16x - 1) + (x³ + 6x² - 11x + 6) + (-2x³ + x² - 6x + 1) Step 2: Combine x³ terms: 6x³ + x³ - 2x³ = 5x³ Step 3: Combine x² terms: -11x² + 6x² + x² = -4x² Step 4: Combine x terms: 16x - 11x - 6x = -x Step 5: Combine constant terms: -1 + 6 + 1 = 6 Step 6: Write the final simplified polynomial: 5x³ - 4x² - x + 6 The answer is 5x³ - 4x² - x + 6.

6. (3x² - 2x + 5) + (2x² + 4x - 3) = ? Answer: 5x² + 2x + 2 Solution: We are adding two polynomials: (3x² - 2x + 5) + (2x² + 4x - 3). Remove the parentheses since we are adding. This gives: 3x² - 2x + 5 + 2x² + 4x - 3.
   Full step-by-step solution

   We are adding two polynomials: (3x² - 2x + 5) + (2x² + 4x - 3). Step 1: Remove the parentheses since we are adding. This gives: 3x² - 2x + 5 + 2x² + 4x - 3. Step 2: Identify like terms. Like terms are terms with the same variable and exponent. - x² terms: 3x² and 2x² - x terms: -2x and 4x - constant terms: 5 and -3 Step 3: Combine the x² terms. 3x² + 2x² = (3 + 2)x² = 5x². Step 4: Combine the x terms. -2x + 4x = (-2 + 4)x = 2x. Step 5: Combine the constant terms. 5 + (-3) = 5 - 3 = 2. Step 6: Write the final simplified polynomial. Putting it all together: 5x² + 2x + 2. Final answer: 5x² + 2x + 2

7. Noah is analyzing the trajectory of a model rocket. The rocket's height above ground at time x seconds is given by the polynomial H(x) = (7x² + 9x - 11) meters. However, due to a tailwind, an additional lift term L(x) = (8x² - 6x + 13) meters must be added to the height. Later, a drag force reduces the height by D(x) = (4x² + 7x - 9) meters. What simplified polynomial represents the rocket's final height F(x) after accounting for both the lift and the drag? Answer: 11x² - 4x + 11 Solution: Write the expression for final height: F(x) = H(x) + L(x) - D(x). Step 2: Substitute the given polynomials: F(x) = (7x² + 9x - 11) + (8x² - 6x + 13) - (4x² + 7x - 9).
   Full step-by-step solution

   Step 1: Write the expression for final height: F(x) = H(x) + L(x) - D(x). Step 2: Substitute the given polynomials: F(x) = (7x² + 9x - 11) + (8x² - 6x + 13) - (4x² + 7x - 9). Step 3: Distribute the subtraction sign: F(x) = 7x² + 9x - 11 + 8x² - 6x + 13 - 4x² - 7x + 9. Step 4: Group like terms: (7x² + 8x² - 4x²) + (9x - 6x - 7x) + (-11 + 13 + 9). Step 5: Simplify each group: (11x²) + (-4x) + (11). Step 6: The final polynomial is 11x² - 4x + 11. The answer is 11x² - 4x + 11.