(x%2B4)(x-2)-(x%5E2%2B11x-9)(x%2B1)%2B5x%5E2.jpeg "(x+5)(x+4)(x-2)-(x^2+11x-9)(x+1)+5x^2")
Simplifying the Expression (x+5)(x+4)(x-2)-(x^2+11x-9)(x+1)+5x^2
This article will guide you through simplifying the given expression:
(x+5)(x+4)(x-2)-(x^2+11x-9)(x+1)+5x^2
Step 1: Expand the Products
We start by expanding the products in the expression using the distributive property (also known as FOIL).
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Expanding (x+5)(x+4): (x+5)(x+4) = x(x+4) + 5(x+4) = x² + 4x + 5x + 20 = x² + 9x + 20
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Expanding (x² + 11x - 9)(x+1): (x² + 11x - 9)(x+1) = x²(x+1) + 11x(x+1) - 9(x+1) = x³ + x² + 11x² + 11x - 9x - 9 = x³ + 12x² + 2x - 9
Now the expression becomes:
(x² + 9x + 20)(x-2) - (x³ + 12x² + 2x - 9)(x+1) + 5x²
Step 2: Expanding the Remaining Products
Let's expand the remaining products:
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Expanding (x² + 9x + 20)(x-2): (x² + 9x + 20)(x-2) = x²(x-2) + 9x(x-2) + 20(x-2) = x³ - 2x² + 9x² - 18x + 20x - 40 = x³ + 7x² + 2x - 40
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Expanding (x³ + 12x² + 2x - 9)(x+1): (x³ + 12x² + 2x - 9)(x+1) = x³(x+1) + 12x²(x+1) + 2x(x+1) - 9(x+1) = x⁴ + x³ + 12x³ + 12x² + 2x² + 2x - 9x - 9 = x⁴ + 13x³ + 14x² - 7x - 9
Now the expression looks like this:
(x³ + 7x² + 2x - 40) - (x⁴ + 13x³ + 14x² - 7x - 9) + 5x²
Step 3: Combining Like Terms
Finally, we combine the like terms:
x³ + 7x² + 2x - 40 - x⁴ - 13x³ - 14x² + 7x + 9 + 5x² =
-x⁴ - 12x³ - 2x² + 9x - 31
Conclusion
Therefore, the simplified form of the given expression (x+5)(x+4)(x-2)-(x^2+11x-9)(x+1)+5x² is -x⁴ - 12x³ - 2x² + 9x - 31.