Polynomial Addition

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Polynomial Addition · Grade 9 · Worksheet 1

1. (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

2. (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

3. Aroha is designing a geometric pattern on a coordinate grid. The pattern consists of a quadrilateral with vertices at (x² + 3x - 1, 0), (0, 5x² - x + 3), (-x² + 2x + 7, 0), and (0, -3x² + 4x - 5). The perimeter of the quadrilateral is found by adding the lengths of its four sides, each represented by a polynomial expression. One side connects (x² + 3x - 1, 0) to (0, 5x² - x + 3) and is represented by the polynomial (5x² + 4x + 4). The second side connects (0, 5x² - x + 3) to (-x² + 2x + 7, 0) and is represented by (7x² - 3x + 10). The third side connects (-x² + 2x + 7, 0) to (0, -3x² + 4x - 5) and is represented by (5x² - 2x + 5). The fourth side connects (0, -3x² + 4x - 5) to (x² + 3x - 1, 0) and is represented by (3x² + x + 3). What is the simplified polynomial expression for the total perimeter of the quadrilateral? Answer: 20x² + 0x + 22 Solution: Side 1: 5x² + 4x + 4 Side 2: 7x² - 3x + 10 Side 3: 5x² - 2x + 5 Side 4: 3x² + x + 3 Total perimeter = (5x² + 4x + 4) + (7x² - 3x + 10) + (5x² - 2x + 5) + (3x² + x + 3) x² terms: 5x² + 7x² + 5x² + 3x² = 20x² x terms: 4x - 3x - 2x + 1x = (4 - 3 - 2 + 1)x = 0x Constant terms: 4 + 10 + 5 + 3 = 22…
   Full step-by-step solution

   Step 1: Write the side lengths in order: Side 1: 5x² + 4x + 4 Side 2: 7x² - 3x + 10 Side 3: 5x² - 2x + 5 Side 4: 3x² + x + 3 Step 2: Add all side lengths: Total perimeter = (5x² + 4x + 4) + (7x² - 3x + 10) + (5x² - 2x + 5) + (3x² + x + 3) Step 3: Group like terms: x² terms: 5x² + 7x² + 5x² + 3x² = 20x² x terms: 4x - 3x - 2x + 1x = (4 - 3 - 2 + 1)x = 0x Constant terms: 4 + 10 + 5 + 3 = 22 Step 4: Write the simplified expression: Total perimeter = 20x² + 0x + 22, which simplifies to 20x² + 22. The answer is 20x² + 22.

4. A mosaic artist, Noah, creates a rectangular panel. The length of the panel is represented by the polynomial (6x² + 11x - 16) cm, and the width is represented by the polynomial (4x² - 6x + 21) cm. A triangular glass inset is placed inside the panel with vertices at coordinates (0, 0), (x² + 6, 0), and (0, 6x - 1). What polynomial expression represents the perimeter of the rectangular panel in simplified form? Answer: 20x² + 10x + 10 Solution: Identify the length and width polynomials. Length L = 6x² + 11x - 16 Width W = 4x² - 6x + 21 Add the length and width.
   Full step-by-step solution

   Step 1: Identify the length and width polynomials. Length L = 6x² + 11x - 16 Width W = 4x² - 6x + 21 Step 2: Add the length and width. L + W = (6x² + 11x - 16) + (4x² - 6x + 21) Group like terms: x² terms: 6x² + 4x² = 10x² x terms: 11x + (-6x) = 5x Constant terms: -16 + 21 = 5 So L + W = 10x² + 5x + 5 Step 3: Apply the perimeter formula for a rectangle: P = 2(L + W). P = 2 × (10x² + 5x + 5) Distribute the 2: P = 20x² + 10x + 10 Step 4: The triangular inset coordinates are extra information and do not affect the perimeter of the rectangle. Final answer: 20x² + 10x + 10

5. 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.

6. (3x² - 5x + 7) + (2x² + 4x - 3) = ? Answer: 5x² - x + 4 Solution: (3x² - 5x + 7) + (2x² + 4x - 3) Since it's addition, the parentheses don't change the signs: 3x² - 5x + 7 + 2x² + 4x - 3 - x² terms: 3x² + 2x² - x terms: -5x + 4x - constant terms: 7 - 3 3x² + 2x² = (3 + 2)x² = 5x² -5x + 4x = (-5 + 4)x = -1x, which is -x 7 - 3 = 4 5x² - x + 4 Final Answer: 5x² -…
   Full step-by-step solution

   Let's add the two polynomials step by step. We have: (3x² - 5x + 7) + (2x² + 4x - 3) **Step 1: Remove parentheses** Since it's addition, the parentheses don't change the signs: 3x² - 5x + 7 + 2x² + 4x - 3 **Step 2: Group like terms** - x² terms: 3x² + 2x² - x terms: -5x + 4x - constant terms: 7 - 3 **Step 3: Combine x² terms** 3x² + 2x² = (3 + 2)x² = 5x² **Step 4: Combine x terms** -5x + 4x = (-5 + 4)x = -1x, which is -x **Step 5: Combine constants** 7 - 3 = 4 **Step 6: Write the final polynomial** 5x² - x + 4 **Final Answer:** 5x² - x + 4