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Factoring and Expanding (x-4)(x+5)
This article explores the process of factoring and expanding the expression (x-4)(x+5). We'll cover the basic concepts involved, provide step-by-step calculations, and demonstrate the application of the distributive property.
Understanding the Expression
The expression (x-4)(x+5) represents the product of two binomial expressions:
- (x-4): This binomial has a variable term 'x' and a constant term '-4'.
- (x+5): This binomial also has a variable term 'x' and a constant term '+5'.
Expanding the Expression
To expand the expression, we apply the distributive property:
Step 1: Multiply the first term of the first binomial by each term of the second binomial.
(x-4)(x+5) = x * (x+5) - 4 * (x+5)
Step 2: Simplify the resulting expressions.
x * (x+5) - 4 * (x+5) = x² + 5x - 4x - 20
Step 3: Combine like terms.
x² + 5x - 4x - 20 = x² + x - 20
Therefore, the expanded form of (x-4)(x+5) is x² + x - 20.
Factoring the Expression
Factoring the expression x² + x - 20 involves finding two binomials that multiply to give the original expression. This is the reverse of expansion.
Step 1: Identify two numbers that multiply to give -20 (the constant term) and add up to 1 (the coefficient of the x term).
These numbers are 5 and -4.
Step 2: Form the two binomials using these numbers.
** (x + 5)(x - 4)**
Therefore, the factored form of x² + x - 20 is (x + 5)(x - 4).
Conclusion
In conclusion, the expression (x-4)(x+5) can be expanded to x² + x - 20 using the distributive property. Conversely, it can be factored into (x + 5)(x - 4). Both forms are equivalent and represent the same algebraic expression. Understanding these operations is essential for solving equations, simplifying expressions, and manipulating algebraic expressions in various mathematical contexts.