This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here are the solutions for the remaining problems in Exercise 8.3. 1. Add the following expressions: 1.7 5m^2 - 4mn + 7n^2; -10n^2 + mn; 7m^2 - 10n^2 Step 1: Group like terms. (5m^2 + 7m^2) + (-4mn + mn) + (7n^2 - 10n^2 - 10n^2) Step 2: Combine like terms. 12m^2 - 3mn - 13n^2 The sum is 12m^2 - 3mn - 13n^2. 1.8 -3x^2 + 3xy - y^2; -4x^2 - 7xy - 10y^2; -3xy + 15y^2 Step 1: Group like terms. (-3x^2 - 4x^2) + (3xy - 7xy - 3xy) + (-y^2 - 10y^2 + 15y^2) Step 2: Combine like terms. -7x^2 - 7xy + 4y^2 The sum is -7x^2 - 7xy + 4y^2. 1.9 5a^2b - 4ab^2 + 2ab; 10a^2b + ab^2 - ab; -6a^2b - 2ab^2 - 3ab Step 1: Group like terms. (5a^2b + 10a^2b - 6a^2b) + (-4ab^2 + ab^2 - 2ab^2) + (2ab - ab - 3ab) Step 2: Combine like terms. 9a^2b - 5ab^2 - 2ab The sum is 9a^2b - 5ab^2 - 2ab. 1.10 8x^3 + 6x^2 - 4x + 5; -x^3 - 2x^2 + 3x - 7 Step 1: Group like terms. (8x^3 - x^3) + (6x^2 - 2x^2) + (-4x + 3x) + (5 - 7) Step 2: Combine like terms. 7x^3 + 4x^2 - x - 2 The sum is 7x^3 + 4x^2 - x - 2. 2. Subtract the second expression from the first expression: 2.1 3a + 7; 5a - 3 Step 1: Write the subtraction. (3a + 7) - (5a - 3) Step 2: Distribute the negative sign and group like terms. 3a + 7 - 5a + 3 = (3a - 5a) + (7 + 3) Step 3: Combine like terms. -2a + 10 The difference is -2a + 10. 2.2 5x + 2y; -9x + 2y Step 1: Write the subtraction. (5x + 2y) - (-9x + 2y) Step 2: Distribute the negative sign and group like terms. 5x + 2y + 9x - 2y = (5x + 9x) + (2y - 2y) Step 3: Combine like terms. 14x + 0y = 14x The difference is 14x. 2.3 3b - 9; 7 - 2b Step 1: Write the subtraction. (3b - 9) - (7 - 2b) Step 2: Distribute the negative sign and group like terms. 3b - 9 - 7 + 2b = (3b + 2b) + (-9 - 7) Step 3: Combine like terms. 5b - 16 The difference is 5b - 16. 2.4 1; 8k + 4 Step 1: Write the subtraction. (1) - (8k + 4) Step 2: Distribute the negative sign and group like terms. 1 - 8k - 4 = -8k + (1 - 4) Step 3: Combine like terms. -8k - 3 The difference is -8k - 3. 2.5 4a + b - 2c; -a + 3b + c Step 1: Write the subtraction. (4a + b - 2c) - (-a + 3b + c) Step 2: Distribute the negative sign and group like terms. 4a + b - 2c + a - 3b - c = (4a + a) + (b - 3b) + (-2c - c) Step 3: Combine like terms. 5a - 2b - 3c The difference is 5a - 2b - 3c. 2.6 3x^2 + 4x - 1; -2x^2 - 3x + 5 Step 1: Write the subtraction. (3x^2 + 4x - 1) - (-2x^2 - 3x + 5) Step 2: Distribute the negative sign and group like terms. 3x^2 + 4x - 1 + 2x^2 + 3x - 5 = (3x^2 + 2x^2) + (4x + 3x) + (-1 - 5) Step 3: Combine like terms. 5x^2 + 7x - 6 The difference is 5x^2 + 7x - 6. 2.7 2a^2 - ab + b^2; 2a^2 + ab - b^2 Step 1: Write the subtraction. (2a^2 - ab + b^2) - (2a^2 + ab - b^2) Step 2: Distribute the negative sign and group like terms. 2a^2 - ab + b^2 - 2a^2 - ab + b^2 = (2a^2 - 2a^2) + (-ab - ab) + (b^2 + b^2) Step 3: Combine like terms. 0a^2 - 2ab + 2b^2 = -2ab + 2b^2 The difference is -2ab + 2b^2. 2.8 5m^2 + 3mn - 5n^2; -5m^2 + 8mn Step 1: Write the subtraction. (5m^2 + 3mn - 5n^2) - (-5m^2 + 8mn) Step 2: Distribute the negative sign and group like terms. 5m^2 + 3mn - 5n^2 + 5m^2 - 8mn = (5m^2 + 5m^2) + (3mn - 8mn) - 5n^2 Step 3: Combine like terms. 10m^2 - 5mn - 5n^2 The difference is 10m^2 - 5mn - 5n^2. 2.9 2a^2b - 4ab^2 + ab; -2a^2b - 4ab^2 + ab Step 1: Write the subtraction. (2a^2b - 4ab^2 + ab) - (-2a^2b - 4ab^2 + ab) Step 2: Distribute the negative sign and group like terms. 2a^2b - 4ab^2 + ab + 2a^2b + 4ab^2 - ab = (2a^2b + 2a^2b) + (-4ab^2 + 4ab^2) + (ab - ab) Step 3: Combine like terms. 4a^2b + 0ab^2 + 0ab = 4a^2b The difference is 4a^2b. 2.10 4y^3 - 7y^2 + 2y - 1; -y^3 + 3y^2 + y - 3 Step 1: Write the subtraction. (4y^3 - 7y^2 + 2y - 1) - (-y^3 + 3y^2 + y - 3) Step 2: Distribute the negative sign and group like terms. 4y^3 - 7y^2 + 2y - 1 + y^3 - 3y^2 - y + 3 = (4y^3 + y^3) + (-7y^2 - 3y^2) + (2y - y) + (-1 + 3) Step 3: Combine like terms. 5y^3 - 10y^2 + y + 2 The difference is 5y^3 - 10y^2 + y + 2. 3 done, 2 left today. You're making progress.