Polynomial Addition

lessonbunny.com

Answer Key & Explanations

Polynomial Addition · Grade 9 · Worksheet 2

1. A tech startup is modeling their quarterly profit using polynomial functions. Their revenue is modeled by R(x) = 4x³ - 2x² + 7x - 3, where x represents thousands of units sold. Their operating costs are modeled by C(x) = x³ + 5x² - 2x + 8. The company's board wants to understand their net profit function. What polynomial represents P(x), their quarterly profit? Answer: 3x³ - 7x² + 9x - 11 Solution: In business mathematics, profit is calculated by subtracting total costs from total revenue. The key is to carefully combine like terms while preserving the structure of the polynomial.
   Full step-by-step solution

   In business mathematics, profit is calculated by subtracting total costs from total revenue. When working with polynomial functions that model real-world scenarios, we perform operations on the expressions while maintaining the mathematical relationships. The key is to carefully combine like terms while preserving the structure of the polynomial.

2. Aroha is analyzing the budget for her small business. Her monthly revenue is modeled by the polynomial \( R(x) = 9x^2 + 14x - 11 \) dollars, and her monthly expenses are modeled by \( E(x) = 4x^2 - 7x + 13 \) dollars, where \( x \) represents the number of units sold in hundreds. Aroha needs to determine her monthly profit, which is revenue minus expenses. What simplified polynomial expression represents her monthly profit, \( P(x) \)? Answer: 5x^2 + 21x - 24 Solution: Profit is revenue minus expenses, so P(x) = R(x) - E(x). Substitute the given expressions: P(x) = (9x^2 + 14x - 11) - (4x^2 - 7x + 13).
   Full step-by-step solution

   Step 1: Profit is revenue minus expenses, so P(x) = R(x) - E(x). Step 2: Substitute the given expressions: P(x) = (9x^2 + 14x - 11) - (4x^2 - 7x + 13). Step 3: Distribute the subtraction sign to each term in the second polynomial: P(x) = 9x^2 + 14x - 11 - 4x^2 + 7x - 13. Step 4: Combine like terms: - x^2 terms: 9x^2 - 4x^2 = 5x^2 - x terms: 14x + 7x = 21x - constant terms: -11 - 13 = -24 Step 5: The simplified polynomial is P(x) = 5x^2 + 21x - 24. The answer is 5x^2 + 21x - 24.

3. (3x² - 2x + 5) + (2x² + 4x - 3) - (x² - 3x + 1) = ? Answer: 4x² + 5x + 1 Solution: (3x² - 2x + 5) + (2x² + 4x - 3) - (x² - 3x + 1) The first two parentheses are added, so they stay the same: 3x² - 2x + 5 + 2x² + 4x - 3 The third parenthesis is subtracted, so we reverse all signs inside it: - (x² - 3x + 1) = -x² + 3x - 1 3x² - 2x + 5 + 2x² + 4x - 3 - x² + 3x - 1 3x² + 2x² - x²…
   Full step-by-step solution

   Let's solve step by step. We have: (3x² - 2x + 5) + (2x² + 4x - 3) - (x² - 3x + 1) --- **Step 1: Remove parentheses carefully** The first two parentheses are added, so they stay the same: 3x² - 2x + 5 + 2x² + 4x - 3 The third parenthesis is subtracted, so we reverse all signs inside it: - (x² - 3x + 1) = -x² + 3x - 1 Now the whole expression is: 3x² - 2x + 5 + 2x² + 4x - 3 - x² + 3x - 1 --- **Step 2: Combine like terms for x² terms** 3x² + 2x² - x² = (3 + 2 - 1)x² = 4x² --- **Step 3: Combine like terms for x terms** -2x + 4x + 3x = (-2 + 4 + 3)x = 5x --- **Step 4: Combine constant terms** 5 - 3 - 1 = (5 - 3) - 1 = 2 - 1 = 1 --- **Step 5: Write the final expression** 4x² + 5x + 1 --- **Final answer:** 4x² + 5x + 1

4. A coordinate grid shows a quadrilateral with vertices at A(2x² - 3x + 1, 0), B(2x² + 4x - 6, 0), C(2x² + 4x - 6, x² + 6x - 1), and D(2x² - 3x + 1, x² + 6x - 1). What polynomial expression represents the perimeter of this quadrilateral? Answer: 6x² + 26x - 16 Solution: Identify the shape. Since A and B share the same y-coordinate (0), AB is horizontal. Since C and D share the same y-coordinate (x² + 6x - 1), CD is horizontal.
   Full step-by-step solution

   Step 1: Identify the shape. Since A and B share the same y-coordinate (0), AB is horizontal. Since C and D share the same y-coordinate (x² + 6x - 1), CD is horizontal. Since A and D share the same x-coordinate (2x² - 3x + 1), AD is vertical. Since B and C share the same x-coordinate (2x² + 4x - 6), BC is vertical. The quadrilateral is a rectangle. Step 2: Find length AB (horizontal). AB = (x-coordinate of B) - (x-coordinate of A) = (2x² + 4x - 6) - (2x² - 3x + 1). Subtract term by term: 2x² - 2x² = 0, 4x - (-3x) = 4x + 3x = 7x, -6 - 1 = -7. So AB = 7x - 7. Step 3: Find length AD (vertical). AD = (y-coordinate of D) - (y-coordinate of A) = (x² + 6x - 1) - 0 = x² + 6x - 1. Step 4: In a rectangle, opposite sides are equal. So CD = AB = 7x - 7, and BC = AD = x² + 6x - 1. Step 5: Perimeter = AB + BC + CD + AD = (7x - 7) + (x² + 6x - 1) + (7x - 7) + (x² + 6x - 1). Combine like terms: x² terms: x² + x² = 2x². x terms: 7x + 6x + 7x + 6x = 26x. Constant terms: -7 - 1 - 7 - 1 = -16. So perimeter = 2x² + 26x - 16. The answer is 2x² + 26x - 16.

5. Hana is designing a mosaic on a rectangular coordinate grid. The mosaic features two overlapping triangular regions. The first triangle has vertices at (0, 0), (4x, 0), and (0, 3x² + 5x). The second triangle has vertices at (0, 0), (2x² - 3x, 0), and (0, 7x - 2). Hana wants to add the polynomial expressions for the lengths of the vertical sides of the two triangles (the sides along the y-axis from the origin to the top vertex) and then subtract the polynomial expression for the horizontal side of the second triangle (the side along the x-axis from the origin to the right vertex). What is the resulting simplified polynomial? Answer: 3x² + 12x - 2x² + 3x - 2 Solution: Identify the vertical side of the first triangle. The top vertex is (0, 3x² + 5x), so the vertical side length is 3x² + 5x. Identify the vertical side of the second triangle.
   Full step-by-step solution

   Step 1: Identify the vertical side of the first triangle. The top vertex is (0, 3x² + 5x), so the vertical side length is 3x² + 5x. Step 2: Identify the vertical side of the second triangle. The top vertex is (0, 7x - 2), so the vertical side length is 7x - 2. Step 3: Add these two vertical side lengths: (3x² + 5x) + (7x - 2) = 3x² + 5x + 7x - 2 = 3x² + 12x - 2. Step 4: Identify the horizontal side of the second triangle. The right vertex on the x-axis is (2x² - 3x, 0), so the horizontal side length is 2x² - 3x. Step 5: Subtract this horizontal side from the sum obtained in Step 3: (3x² + 12x - 2) - (2x² - 3x). Step 6: Distribute the subtraction: 3x² + 12x - 2 - 2x² + 3x. Step 7: Combine like terms: 3x² - 2x² = x²; 12x + 3x = 15x; constant term is -2. Step 8: The final simplified polynomial is x² + 15x - 2. The answer is x² + 15x - 2.

6. (5x³ - 10x² + 15x - 20) + (10x³ + 5x² - 20x + 25) - (5x³ - 15x² + 10x - 5) = ? Answer: 10x³ + 10x² - 15x + 10 Solution: Write the expression: (5x³ - 10x² + 15x - 20) + (10x³ + 5x² - 20x + 25) - (5x³ - 15x² + 10x - 5) Distribute the subtraction sign to the third polynomial: (5x³ - 10x² + 15x - 20) + (10x³ + 5x² - 20x + 25) + (-5x³ + 15x² - 10x + 5) Combine x³ terms: 5x³ + 10x³ - 5x³ = 10x³ Combine x² terms: -10x²…
   Full step-by-step solution

   Step 1: Write the expression: (5x³ - 10x² + 15x - 20) + (10x³ + 5x² - 20x + 25) - (5x³ - 15x² + 10x - 5) Step 2: Distribute the subtraction sign to the third polynomial: (5x³ - 10x² + 15x - 20) + (10x³ + 5x² - 20x + 25) + (-5x³ + 15x² - 10x + 5) Step 3: Combine x³ terms: 5x³ + 10x³ - 5x³ = 10x³ Step 4: Combine x² terms: -10x² + 5x² + 15x² = 10x² Step 5: Combine x terms: 15x - 20x - 10x = -15x Step 6: Combine constant terms: -20 + 25 + 5 = 10 Step 7: Write the final answer: 10x³ + 10x² - 15x + 10

7. (3x² - 5x + 7) + (2x² + 4x - 9) - (x² - 3x + 2) = ? Answer: 4x² + 2x - 4 Solution: (3x² - 5x + 7) + (2x² + 4x - 9) - (x² - 3x + 2) The first two parentheses are added, so they stay as they are: 3x² - 5x + 7 + 2x² + 4x - 9 The third parenthesis is subtracted, so we reverse all its signs: - (x² - 3x + 2) = -x² + 3x - 2 3x² - 5x + 7 + 2x² + 4x - 9 - x² + 3x - 2 3x² + 2x² - x² =…
   Full step-by-step solution

   Let's solve step by step. We have: (3x² - 5x + 7) + (2x² + 4x - 9) - (x² - 3x + 2) --- **Step 1: Remove parentheses carefully** The first two parentheses are added, so they stay as they are: 3x² - 5x + 7 + 2x² + 4x - 9 The third parenthesis is subtracted, so we reverse all its signs: - (x² - 3x + 2) = -x² + 3x - 2 So the whole expression becomes: 3x² - 5x + 7 + 2x² + 4x - 9 - x² + 3x - 2 --- **Step 2: Combine like terms (x² terms)** 3x² + 2x² - x² = (3 + 2 - 1)x² = 4x² --- **Step 3: Combine x terms** -5x + 4x + 3x = (-5 + 4 + 3)x = 2x --- **Step 4: Combine constant terms** 7 - 9 - 2 = (7 - 9) - 2 = (-2) - 2 = -4 --- **Step 5: Write final expression** 4x² + 2x - 4 --- **Final answer:** 4x² + 2x - 4