Math Problem Statement

-2(-x)^5(6x-9)^4((-x)^2+1))^2

Solution

Let's break down and simplify the expression you provided step by step:

2(x)5(6x9)4((x)2+1)2-2(-x)^5(6x-9)^4((-x)^2 + 1)^2

Step 1: Simplify each component

(x)5(-x)^5

Since the exponent is odd, this expression simplifies as follows:

(x)5=x5(-x)^5 = -x^5

(6x9)4(6x - 9)^4

We have already expanded (6x9)4(6x - 9)^4 in the previous step. The result was:

(6x9)4=1296x47776x3+17496x217496x+6561(6x - 9)^4 = 1296x^4 - 7776x^3 + 17496x^2 - 17496x + 6561

(x)2+1(-x)^2 + 1

Here, (x)2(-x)^2 simplifies to x2x^2, so:

(x)2+1=x2+1(-x)^2 + 1 = x^2 + 1

Now, squaring this expression gives:

(x2+1)2=x4+2x2+1(x^2 + 1)^2 = x^4 + 2x^2 + 1

Step 2: Combine the components

Now that we've simplified the individual components, we can substitute them back into the original expression:

2(x)5(6x9)4((x)2+1)2-2(-x)^5(6x - 9)^4((-x)^2 + 1)^2

Substituting the simplifications:

2(x5)(1296x47776x3+17496x217496x+6561)(x4+2x2+1)-2(-x^5)(1296x^4 - 7776x^3 + 17496x^2 - 17496x + 6561)(x^4 + 2x^2 + 1)

Step 3: Simplify the negative terms

We already have x5-x^5 inside the expression. Multiplying by the negative 2 will simplify as:

2(x5)=2x5-2(-x^5) = 2x^5

So the expression becomes:

[ 2

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Math Problem Analysis

Mathematical Concepts

Algebra
Polynomial Expansion
Exponent Rules
Quadratic Functions

Formulas

(-x)^n = (-1)^n * x^n for odd and even powers
(a-b)^n formula for binomial expansion
(a+b)^2 = a^2 + 2ab + b^2

Theorems

Exponent Laws
Polynomial Expansion Theorem

Suitable Grade Level

Grades 10-12