Math Problem Statement

Write \(\frac{x^2 - 3x - 4}{x^3 - 4x^2 - x + 4}\) in simplest form.

Solution

We need to simplify the given rational expression:

x23x4x34x2x+4\frac{x^2 - 3x - 4}{x^3 - 4x^2 - x + 4}

Step 1: Factor the numerator

The numerator is x23x4x^2 - 3x - 4. We will factor this quadratic expression. We need to find two numbers that multiply to 4-4 and add to 3-3. These numbers are 4-4 and 11. Thus, we can factor the numerator as:

x23x4=(x4)(x+1)x^2 - 3x - 4 = (x - 4)(x + 1)

Step 2: Factor the denominator

The denominator is x34x2x+4x^3 - 4x^2 - x + 4. We will factor this cubic expression by grouping.

x34x2x+4=(x34x2)(x4)x^3 - 4x^2 - x + 4 = (x^3 - 4x^2) - (x - 4)

Factor out x2x^2 from the first group and 1-1 from the second group:

x2(x4)1(x4)=(x21)(x4)x^2(x - 4) - 1(x - 4) = (x^2 - 1)(x - 4)

Now, factor x21x^2 - 1 as a difference of squares:

x21=(x1)(x+1)x^2 - 1 = (x - 1)(x + 1)

Thus, the denominator becomes:

(x21)(x4)=(x1)(x+1)(x4)(x^2 - 1)(x - 4) = (x - 1)(x + 1)(x - 4)

Step 3: Simplify the expression

Now we have the following expression:

(x4)(x+1)(x1)(x+1)(x4)\frac{(x - 4)(x + 1)}{(x - 1)(x + 1)(x - 4)}

We can cancel out the common factors (x4)(x - 4) and (x+1)(x + 1):

1x1\frac{1}{x - 1}

Final Answer:

The simplest form of the expression is:

1x1\frac{1}{x - 1}

Thus, the correct answer is c.

Would you like further clarification or details?

Here are 5 related questions:

  1. How do you factor quadratic expressions in general?
  2. What is the method of factoring by grouping?
  3. How do you simplify rational expressions?
  4. Can you always cancel terms in the numerator and denominator?
  5. What are difference of squares, and how are they factored?

Tip: When simplifying rational expressions, always check for common factors in both the numerator and the denominator.

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Math Problem Analysis

Mathematical Concepts

Algebra
Factoring Polynomials
Rational Expressions

Formulas

Difference of Squares: \(a^2 - b^2 = (a - b)(a + b)\)

Theorems

Difference of Squares

Suitable Grade Level

Grades 9-12