-(2a%2B4b%2B7).jpeg "(2a-4b+7) (2a+4b+7)")
Expanding the Expression: (2a-4b+7)(2a+4b+7)
This expression represents the product of two trinomials, and we can expand it using the distributive property or the FOIL method.
Understanding the Structure
Before we start expanding, it's helpful to understand the structure of the expression:
- (2a-4b+7): This is the first trinomial.
- (2a+4b+7): This is the second trinomial.
Expanding using the Distributive Property
The distributive property states that multiplying a sum by a number is the same as multiplying each term of the sum by that number. We can apply this property multiple times to expand the expression:
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Distribute the first term of the first trinomial (2a):
- 2a * (2a + 4b + 7) = 4a² + 8ab + 14a
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Distribute the second term of the first trinomial (-4b):
- -4b * (2a + 4b + 7) = -8ab - 16b² - 28b
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Distribute the third term of the first trinomial (7):
- 7 * (2a + 4b + 7) = 14a + 28b + 49
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Combine all the terms:
- 4a² + 8ab + 14a - 8ab - 16b² - 28b + 14a + 28b + 49
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Simplify by combining like terms:
- 4a² - 16b² + 28a + 49
Expanding using the FOIL Method
The FOIL method is a mnemonic for remembering the order in which to multiply the terms of two binomials:
- First: Multiply the first terms of each binomial.
- Outer: Multiply the outer terms of each binomial.
- Inner: Multiply the inner terms of each binomial.
- Last: Multiply the last terms of each binomial.
We can apply this method to our expression by treating it as a product of two binomials:
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First: (2a) * (2a) = 4a²
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Outer: (2a) * (7) = 14a
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Inner: (-4b) * (2a) = -8ab
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Last: (-4b) * (7) = -28b
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Repeat the process for the remaining terms:
- (2a) * (4b) = 8ab
- (2a) * (7) = 14a
- (-4b) * (4b) = -16b²
- (-4b) * (7) = -28b
- (7) * (2a) = 14a
- (7) * (4b) = 28b
- (7) * (7) = 49
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Combine all the terms:
- 4a² + 14a - 8ab - 28b + 8ab + 14a - 16b² - 28b + 14a + 28b + 49
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Simplify by combining like terms:
- 4a² - 16b² + 28a + 49
Conclusion
Both methods lead to the same simplified expression: 4a² - 16b² + 28a + 49. This is the expanded form of the original expression (2a-4b+7)(2a+4b+7).