(x%2B5)-(x-3)(x-2).jpeg "(x-4)(x+5) (x-3)(x-2)")
Expanding and Simplifying the Expression: (x-4)(x+5)(x-3)(x-2)
This article explores the process of expanding and simplifying the expression (x-4)(x+5)(x-3)(x-2). We will utilize the distributive property and combine like terms to arrive at a simplified polynomial.
Step 1: Expanding the first two factors
We begin by expanding the first two factors, (x-4)(x+5), using the distributive property (also known as FOIL):
(x-4)(x+5) = x(x+5) - 4(x+5) = x² + 5x - 4x - 20 = x² + x - 20
Step 2: Expanding the last two factors
Next, we expand the last two factors, (x-3)(x-2) in the same manner:
(x-3)(x-2) = x(x-2) - 3(x-2) = x² - 2x - 3x + 6 = x² - 5x + 6
Step 3: Expanding the two simplified expressions
Now, we have two simplified expressions: x² + x - 20 and x² - 5x + 6. We will multiply these expressions using the distributive property again:
(x² + x - 20)(x² - 5x + 6) = x²(x² - 5x + 6) + x(x² - 5x + 6) - 20(x² - 5x + 6)
Expanding each term:
= x⁴ - 5x³ + 6x² + x³ - 5x² + 6x - 20x² + 100x - 120
Step 4: Combining like terms
Finally, we combine the terms with the same powers of x:
= x⁴ - 4x³ - 19x² + 106x - 120
Conclusion
Therefore, the simplified form of the expression (x-4)(x+5)(x-3)(x-2) is x⁴ - 4x³ - 19x² + 106x - 120.