If P = Q - 4, then the value of P³ - Q³ + 12 PQ + 64 is 0
Step-by-step explanation:
To find the value of the polynomial
- Substitute P by Q - 4 in the polynomial
- Solve the brackets
- Add the like terms
∵ P = Q - 4
∵ P³ - Q³ + 12 PQ + 64
- Substitute P by Q - 4 to find P³
∴ P³ = (Q - 4)³
- Distribute (Q - 4)³ into (Q - 4)(Q - 4)(Q - 4) and multiply them
∵ (Q - 4)(Q - 4) = Q(Q) + Q(-4) - 4(Q) - 4(-4) = Q² - 4Q - 4Q + 16
- Add like terms
∴ (Q - 4)(Q - 4) = Q² - 8Q + 16
∵ (Q² - 8Q + 16)(Q - 4) = Q²(Q) - Q²(4) - 8Q(Q) - 8Q(-4) + 16(Q) - 16(4)
∴ (Q² - 8Q + 16)(Q - 4) = Q³ - 4Q² - 8Q² + 32Q + 16Q - 64
- Add like terms
∴ (Q² - 8Q + 16)(Q - 4) = Q³ - 12Q² + 48Q - 64
∴ (Q - 4)(Q - 4)(Q - 4) = Q³ - 12Q² + 48Q - 64
∴ (Q - 4)³ = Q³ - 12Q² + 48Q - 64
∴ P³ = Q³ - 12Q² + 48Q - 64
- Substitute P by Q - 4 to find 12 PQ
∵ 12 PQ = 12Q(Q - 4) = 12(Q)(Q) - 12(4)
∴ 12 PQ = 12Q² - 48Q
- Substitute P³ by (Q³ - 12Q² + 48Q - 64) and 12 PQ by (12Q² - 48Q)
in the polynomial
∵ P³ - Q³ + 12 PQ + 64 = (Q³ - 12Q² + 48Q - 64) - Q³ + (12Q² - 48Q) + 64
- Now collect the like terms
∴ P³ - Q³ + 12 PQ + 64 = (Q³ - Q³) + (-12Q² + 12Q²) + (48Q - 48Q) + (-64 + 64)
∴ P³ - Q³ + 12 PQ + 64 = 0 + 0 + 0 + 0
∴ P³ - Q³ + 12 PQ + 64 = 0
If P = Q - 4, then the value of P³ - Q³ + 12 PQ + 64 is 0
Learn more:
You can learn more about the polynomials in brainly.com/question/9184197
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