Exercise 8.3: Add the following expressions and subtract the second expression from the first.

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Here are the solutions for the remaining problems in Exercise 8.3. 1. Add the following expressions: 1.7 5m² − 4mn + 7n²; −10n² + mn; 7m² − 10n² Step 1: Group like terms. (5m² + 7m²) + (−4mn + mn) + (7n² − 10n² − 10n²) Step 2: Combine like terms. 12m² − 3mn − 13n² The sum is 12m² − 3mn − 13n². 1.8 −3x² + 3xy − y²; −4x² − 7xy − 10y²; −3xy + 15y² Step 1: Group like terms. (−3x² − 4x²) + (3xy − 7xy − 3xy) + (−y² − 10y² + 15y²) Step 2: Combine like terms. −7x² − 7xy + 4y² The sum is −7x² − 7xy + 4y². 1.9 5a²b − 4ab² + 2ab; 10a²b + ab² − ab; −6a²b − 2ab² − 3ab Step 1: Group like terms. (5a²b + 10a²b − 6a²b) + (−4ab² + ab² − 2ab²) + (2ab − ab − 3ab) Step 2: Combine like terms. 9a²b − 5ab² − 2ab The sum is 9a²b − 5ab² − 2ab. 1.10 8x³ + 6x² − 4x + 5; −x³ − 2x² + 3x − 7 Step 1: Group like terms. (8x³ − x³) + (6x² − 2x²) + (−4x + 3x) + (5 − 7) Step 2: Combine like terms. 7x³ + 4x² − x − 2 The sum is 7x³ + 4x² − 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 3x² + 4x − 1; −2x² − 3x + 5 Step 1: Write the subtraction. (3x² + 4x − 1) − (−2x² − 3x + 5) Step 2: Distribute the negative sign and group like terms. 3x² + 4x − 1 + 2x² + 3x − 5 = (3x² + 2x²) + (4x + 3x) + (−1 − 5) Step 3: Combine like terms. 5x² + 7x − 6 The difference is 5x² + 7x − 6. 2.7 2a² − ab + b²; 2a² + ab − b² Step 1: Write the subtraction. (2a² − ab + b²) − (2a² + ab − b²) Step 2: Distribute the negative sign and group like terms. 2a² − ab + b² − 2a² − ab + b² = (2a² − 2a²) + (−ab − ab) + (b² + b²) Step 3: Combine like terms. 0a² − 2ab + 2b² = −2ab + 2b² The difference is −2ab + 2b². 2.8 5m² + 3mn − 5n²; −5m² + 8mn Step 1: Write the subtraction. (5m² + 3mn − 5n²) − (−5m² + 8mn) Step 2: Distribute the negative sign and group like terms. 5m² + 3mn − 5n² + 5m² − 8mn = (5m² + 5m²) + (3mn − 8mn) − 5n² Step 3: Combine like terms. 10m² − 5mn − 5n² The difference is 10m² − 5mn − 5n². 2.9 2a²b − 4ab² + ab; −2a²b − 4ab² + ab Step 1: Write the subtraction. (2a²b − 4ab² + ab) − (−2a²b − 4ab² + ab) Step 2: Distribute the negative sign and group like terms. 2a²b − 4ab² + ab + 2a²b + 4ab² − ab = (2a²b + 2a²b) + (−4ab² + 4ab²) + (ab − ab) Step 3: Combine like terms. 4a²b + 0ab² + 0ab = 4a²b The difference is 4a²b. 2.10 4y³ − 7y² + 2y − 1; −y³ + 3y² + y − 3 Step 1: Write the subtraction. (4y³ − 7y² + 2y − 1) − (−y³ + 3y² + y − 3) Step 2: Distribute the negative sign and group like terms. 4y³ − 7y² + 2y − 1 + y³ − 3y² − y + 3 = (4y³ + y³) + (−7y² − 3y²) + (2y − y) + (−1 + 3) Step 3: Combine like terms. 5y³ − 10y² + y + 2 The difference is 5y³ − 10y² + y + 2.

Exercise 8.3: Add the following expressions and subtract the second expression from the first.

Step-by-step solution

Part 1.7· 2 steps

Group like terms.

Combine like terms.

Part 1.8· 2 steps

Group like terms.

Combine like terms.

Part 1.9· 2 steps

Group like terms.

Combine like terms.

Part 1.10· 2 steps

Group like terms.

Combine like terms.

Part 2.1· 3 steps

Write the subtraction.

Distribute the negative sign and group like terms.

Combine like terms.

Part 2.2· 3 steps

Write the subtraction.

Distribute the negative sign and group like terms.

Combine like terms.

Part 2.3· 3 steps

Write the subtraction.

Distribute the negative sign and group like terms.

Combine like terms.

Part 2.4· 3 steps

Write the subtraction.

Distribute the negative sign and group like terms.

Combine like terms.

Part 2.5· 3 steps

Write the subtraction.

Distribute the negative sign and group like terms.

Combine like terms.

Part 2.6· 3 steps

Write the subtraction.

Distribute the negative sign and group like terms.

Combine like terms.

Part 2.7· 3 steps

Write the subtraction.

Distribute the negative sign and group like terms.

Combine like terms.

Part 2.8· 3 steps

Write the subtraction.

Distribute the negative sign and group like terms.

Combine like terms.

Part 2.9· 3 steps

Write the subtraction.

Distribute the negative sign and group like terms.

Combine like terms.

Part 2.10· 3 steps

Write the subtraction.

Distribute the negative sign and group like terms.

Combine like terms.

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