Math Problem Statement

Simplify the expression: (4x^2 + 4x) / (x^4 + x^3)

Solution

The given expression is:

4x2+4xx4+x3\frac{4x^2 + 4x}{x^4 + x^3}

Step 1: Factor the numerator and the denominator.

  • Numerator: 4x2+4x4x^2 + 4x

    • Factor out 4x4x: 4x(x+1)4x(x + 1)
  • Denominator: x4+x3x^4 + x^3

    • Factor out x3x^3: x3(x+1)x^3(x + 1)

Step 2: Simplify the expression.

Now the expression looks like this:

4x(x+1)x3(x+1)\frac{4x(x + 1)}{x^3(x + 1)}

Since (x+1)(x + 1) appears in both the numerator and the denominator, you can cancel (x+1)(x + 1) (as long as x1x \neq -1):

4xx3\frac{4x}{x^3}

Step 3: Simplify further.

Now simplify the remaining expression:

4xx3=4x2\frac{4x}{x^3} = \frac{4}{x^2}

So, the simplified expression is:

4x2\frac{4}{x^2}

Final Answer:

The completely simplified expression is 4x2\frac{4}{x^2}.

Do you have any questions or want more details on any of the steps?


Here are 5 related questions to explore further:

  1. How would the result change if the expression was 4x2+2xx4+x3\frac{4x^2 + 2x}{x^4 + x^3}?
  2. What happens if x=1x = -1? Can the expression still be simplified?
  3. Can you apply the same factoring technique to other polynomial expressions?
  4. How do you simplify expressions with higher-degree polynomials?
  5. What other common factorization patterns are useful in simplifying rational expressions?

Tip: Always check for common factors in both the numerator and the denominator when simplifying rational expressions.

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Math Problem Analysis

Mathematical Concepts

Algebra
Simplifying Rational Expressions
Factoring

Formulas

Factoring Polynomials
Cancelling common terms in rational expressions

Theorems

Fundamental Theorem of Algebra

Suitable Grade Level

Grades 8-10