(x%2B3)(2-x)(1-x)%5E(2))%2F(x%5E(4)(x%2B6)(x-9)(2x%5E(2)%2B4x%2B9)).jpeg "((2x-1)(x+3)(2-x)(1-x)^(2))/(x^(4)(x+6)(x-9)(2x^(2)+4x+9))")
Simplifying Algebraic Expressions: A Step-by-Step Guide
In this article, we will explore the simplification of a complex algebraic expression:
$\frac{(2x-1)(x+3)(2-x)(1-x)^2}{x^4(x+6)(x-9)(2x^2+4x+9)}$
Step 1: Simplify the Numerator
Let's start by simplifying the numerator:
$(2x-1)(x+3)(2-x)(1-x)^2$
We can start by expanding the product of the first two terms:
$(2x-1)(x+3) = 2x^2 + 5x - 3$
Next, we can multiply the result by $(2-x)$:
$(2x^2 + 5x - 3)(2-x) = 4x^2 - 3x - 6 - 2x^2 - 5x + 3x + 3$
Simplifying the expression, we get:
$x^2 - 5x - 3$
Now, we can multiply the result by $(1-x)^2$:
$(x^2 - 5x - 3)(1-x)^2 = x^2 - 5x - 3 - 2x^2 + 10x + 6 + x^2 - 5x - 3$
Simplifying the expression, we get:
$-x^2 + 2$
Step 2: Simplify the Denominator
Now, let's simplify the denominator:
$x^4(x+6)(x-9)(2x^2+4x+9)$
We can start by expanding the product of the first two terms:
$x^4(x+6) = x^5 + 6x^4$
Next, we can multiply the result by $(x-9)$:
$(x^5 + 6x^4)(x-9) = x^6 - 9x^5 + 6x^5 - 54x^4$
Simplifying the expression, we get:
$x^6 - 3x^5 - 54x^4$
Finally, we can multiply the result by $(2x^2+4x+9)$:
$(x^6 - 3x^5 - 54x^4)(2x^2+4x+9) = 2x^8 - 6x^7 - 108x^6 + 8x^6 + 12x^5 + 36x^4 + 18x^2 - 72x - 486x^3$
Simplifying the expression, we get:
$2x^8 - 6x^7 - 100x^6 + 12x^5 + 54x^4 - 486x^3 - 72x - 486$
Step 3: Simplify the Expression
Now that we have simplified both the numerator and the denominator, we can simplify the entire expression:
$\frac{-x^2 + 2}{2x^8 - 6x^7 - 100x^6 + 12x^5 + 54x^4 - 486x^3 - 72x - 486}$
Simplifying the expression, we get:
$\frac{x^2 - 2}{2x^8 - 6x^7 - 100x^6 + 12x^5 + 54x^4 - 486x^3 - 72x - 486}$
And that's it! We have successfully simplified the complex algebraic expression.
