%5E2.jpeg "(4t^3-5)^2")
Expanding the Square of a Trinomial: (4t^3 - 5)^2
The expression (4t^3 - 5)^2 represents the square of a trinomial. To expand it, we can utilize the pattern of squaring a binomial, or simply multiply the expression by itself.
Using the Square of a Binomial Pattern
Recall that the square of a binomial (a - b)^2 expands as: (a - b)^2 = a^2 - 2ab + b^2. Applying this pattern to our expression:
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Identify 'a' and 'b':
- a = 4t^3
- b = 5
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Apply the pattern:
- (4t^3 - 5)^2 = (4t^3)^2 - 2(4t^3)(5) + (5)^2
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Simplify:
- (4t^3 - 5)^2 = 16t^6 - 40t^3 + 25
Expanding by Direct Multiplication
Alternatively, we can simply multiply the expression by itself:
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Write the expression twice:
- (4t^3 - 5)^2 = (4t^3 - 5)(4t^3 - 5)
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Use the distributive property (FOIL method):
- (4t^3 - 5)(4t^3 - 5) = (4t^3)(4t^3) + (4t^3)(-5) + (-5)(4t^3) + (-5)(-5)
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Simplify:
- (4t^3 - 5)^2 = 16t^6 - 40t^3 + 25
Conclusion
Both methods result in the same expanded expression: 16t^6 - 40t^3 + 25. The choice of method depends on personal preference and the complexity of the expression. The pattern approach can be quicker for simpler expressions, while direct multiplication can be more helpful for more complex cases.