Practice: Identities

Answer: a² + 2ab + b²
Step-by-step Explanation:
1. We use the identity (a + b)² = a² + 2ab + b².
2. This means we multiply (a + b) by itself: (a + b) × (a + b).
3. First, multiply a by both terms in the second bracket: a × a = a², a × b = ab.
4. Next, multiply b by both terms: b × a = ab, b × b = b².
5. Add all the terms: a² + ab + ab + b².
6. Combine like terms: a² + 2ab + b².
7. So, (a + b)² = a² + 2ab + b².

Answer: (x + 3)(x - 3)
Step-by-step Explanation:
1. We use the identity a² - b² = (a + b)(a - b).
2. Here, x² - 9 can be written as x² - 3².
3. So, a = x and b = 3.
4. Substitute into the identity: (x + 3)(x - 3).
5. Therefore, x² - 9 = (x + 3)(x - 3).

Answer: a² - 2ab + b²
Step-by-step Explanation:
1. We use the identity (a - b)² = a² - 2ab + b².
2. This means we multiply (a - b) by itself: (a - b) × (a - b).
3. First, multiply a by both terms: a × a = a², a × (-b) = -ab.
4. Next, multiply -b by both terms: -b × a = -ab, -b × -b = b².
5. Add all the terms: a² - ab - ab + b².
6. Combine like terms: a² - 2ab + b².
7. So, (a - b)² = a² - 2ab + b².

Answer: (x + 3)²
Step-by-step Explanation:
1. Notice that x² + 6x + 9 looks like the expansion of (x + b)².
2. Let's compare: (x + 3)² = x² + 2×x×3 + 3² = x² + 6x + 9.
3. So, x² + 6x + 9 = (x + 3)².

Answer: 4x² - 20xy + 25y²
Step-by-step Explanation:
1. Use the identity (a - b)² = a² - 2ab + b².
2. Here, a = 2x and b = 5y.
3. (2x - 5y)² = (2x)² - 2×2x×5y + (5y)².
4. Calculate each part: (2x)² = 4x², 2×2x×5y = 20xy, (5y)² = 25y².
5. Put it together: 4x² - 20xy + 25y².

Answer: (a + b)²
Step-by-step Explanation:
1. This is the expansion of (a + b)².
2. (a + b)² = a² + 2ab + b².
3. So, a² + 2ab + b² = (a + b)².

Answer: x² + 8x + 16
Step-by-step Explanation:
1. Use the identity (a + b)² = a² + 2ab + b².
2. Here, a = x and b = 4.
3. (x + 4)² = x² + 2×x×4 + 4².
4. Calculate: x² + 8x + 16.

Answer: a² + b² + c² + 2ab + 2bc + 2ca
Step-by-step Explanation:
1. Use the identity (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca.
2. Expand (a + b + c) × (a + b + c).
3. Multiply each term in the first bracket by each in the second.
4. Collect like terms: a², b², c², ab, bc, ca.
5. Add: a² + b² + c² + ab + ab + bc + bc + ca + ca.
6. Combine: a² + b² + c² + 2ab + 2bc + 2ca.

Answer: (x - y)²
Step-by-step Explanation:
1. This is the expansion of (x - y)².
2. (x - y)² = x² - 2xy + y².
3. So, x² - 2xy + y² = (x - y)².

Answer: a³ + 3a²b + 3ab² + b³
Step-by-step Explanation:
1. Use the identity (a + b)³ = a³ + 3a²b + 3ab² + b³.
2. (a + b)³ = (a + b) × (a + b) × (a + b).
3. First, expand (a + b) × (a + b) = a² + 2ab + b².
4. Now, multiply (a + b) by (a² + 2ab + b²):
- a × a² = a³ - a × 2ab = 2a²b - a × b² = ab² - b × a² = a²b - b × 2ab = 2ab² - b × b² = b³ 5. Add all terms: a³ + 2a²b + ab² + a²b + 2ab² + b³.
6. Combine like terms: a³ + 3a²b + 3ab² + b³.

Answer: (x + 3)(x² - 3x + 9)
Step-by-step Explanation:
1. Use the identity a³ + b³ = (a + b)(a² - ab + b²).
2. Here, x³ + 27 = x³ + 3³.
3. So, a = x and b = 3.
4. Substitute into the identity: (x + 3)(x² - 3x + 9).

Answer: 8x³ - 36x²y + 54xy² - 27y³
Step-by-step Explanation:
1. Use the identity (a - b)³ = a³ - 3a²b + 3ab² - b³.
2. Here, a = 2x and b = 3y.
3. (2x - 3y)³ = (2x)³ - 3×(2x)²×3y + 3×2x×(3y)² - (3y)³.
4. Calculate each part: (2x)³ = 8x³, (2x)² = 4x², 3×4x²×3y = 36x²y, (3y)² = 9y², 3×2x×9y² = 54xy², (3y)³ = 27y³.
5. Put it together: 8x³ - 36x²y + 54xy² - 27y³.

Answer: (a - b)(a² + ab + b²)
Step-by-step Explanation:
1. Use the identity a³ - b³ = (a - b)(a² + ab + b²).
2. Substitute a and b into the identity.

Answer: x² + 4xy + 9z² + 4y² + 12yz + 12xz
Step-by-step Explanation:
1. Use the identity (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca.
2. Here, a = x, b = 2y, c = 3z.
3. (x + 2y + 3z)² = x² + (2y)² + (3z)² + 2×x×2y + 2×2y×3z + 2×3z×x.
4. Calculate: x² + 4y² + 9z² + 4xy + 12yz + 6xz.

Answer: True
Step-by-step Explanation:
1. Expand (a + b + c) × (a + b + c).
2. Multiply each term in the first bracket by each in the second.
3. Collect like terms: a², b², c², ab, bc, ca.
4. Add: a² + b² + c² + ab + ab + bc + bc + ca + ca.
5. Combine: a² + b² + c² + 2ab + 2bc + 2ca.

Answer: True
Step-by-step Explanation:
1. Use the identity a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca).
2. If a + b + c = 0, then the right side becomes 0.
3. So, a³ + b³ + c³ - 3abc = 0 ⇒ a³ + b³ + c³ = 3abc.

Answer: (x² + 4)(x + 2)(x - 2)
Step-by-step Explanation:
1. x⁴ - 16 = (x²)² - 4².
2. Use the identity a² - b² = (a + b)(a - b).
3. Here, a = x², b = 4.
4. So, (x² + 4)(x² - 4).
5. Now, factor x² - 4 again: (x + 2)(x - 2). 6. Final answer: (x² + 4)(x + 2)(x - 2).

Answer: a³ - 3a²b + 3ab² - b³
Step-by-step Explanation:
1. Use the identity (a - b)³ = a³ - 3a²b + 3ab² - b³.
2. (a - b)³ = (a - b) × (a - b) × (a - b).
3. First, expand (a - b) × (a - b) = a² - 2ab + b².
4. Now, multiply (a - b) by (a² - 2ab + b²):
- a × a² = a³ - a × -2ab = -2a²b - a × b² = ab² - -b × a² = -a²b - -b × -2ab = 2ab² - -b × b² = -b³ 5. Add all terms: a³ - 2a²b + ab² - a²b + 2ab² - b³.
6. Combine like terms: a³ - 3a²b + 3ab² - b³.

Answer: (a² + b²)(a + b)(a - b)
Step-by-step Explanation:
1. a⁴ - b⁴ = (a²)² - (b²)².
2. Use the identity a² - b² = (a + b)(a - b).
3. Here, a = a², b = b².
4. So, (a² + b²)(a² - b²).
5. Now, factor a² - b² again: (a + b)(a - b). 6. Final answer: (a² + b²)(a + b)(a - b).

Answer: 2
Step-by-step Explanation:
1. We are given x + 1/x = 2.
2. Use the identity (a + b)³ = a³ + b³ + 3ab(a + b).
3. Let a = x, b = 1/x.
4. So, (x + 1/x)³ = x³ + 1/x³ + 3x × 1/x × (x + 1/x).
5. x × 1/x = 1, so 3 × 1 × (x + 1/x) = 3 × 2 = 6.
6. (x + 1/x)³ = 8 (since 2³ = 8).
7. So, 8 = x³ + 1/x³ + 6 ⇒ x³ + 1/x³ = 2.