Question

Which rational expression is in simplifi ed form? (A) \(\frac{x^2-x-6}{x^2+3 x+2}\) (B) \(\frac{x^2+6 x+8}{x^2+2 x-3}\) (C) \(\frac{x^2-6 x+9}{x^2-2 x-3}\) (D) \(\frac{x^2+3 x-4}{x^2+x-2}\)

Short answer

Option B, \(\frac{x^2+6x+8}{x^2+2x-3}\) is the rational expression in simplified form.

Step-by-step solution

Simplifying Option A

Factoring the numerator for option A, \(x^2-x-6\) we get \((x-3)(x+2)\). Factoring the denominator \(x^2+3x+2\), we get \((x+2)(x+1)\). Although both numerator and denominator are factored, they have common factors of \((x+2)\), so the expression isn't simplified.

Simplifying Option B

Factoring the numerator for option B, \(x^2+6x+8\) we get \((x+2)(x+4)\). Factoring the denominator \(x^2+2x-3\), it results in \((x+3)(x-1)\). No common factors between the numerator and denominator. Therefore, option B is in simplified form.

Simplifying Options C and D

Even if we've found a correct answer in step 2, let's continue factoring for practice. Factoring the numerator for option C, \(x^2-6x+9\) we get \((x-3)^2\). Factoring the denominator \(x^2-2x-3\), we get \((x-3)(x+1)\). There is a common factor of \(x-3\), so option C isn't in simplified form. Looking at option D, the same process will reveal that it also isn't in simplified form since the numerator \(x^2+3x-4\) factors as \((x+4)(x-1)\) and the denominator \(x^2+x-2\) factors as \((x+2)(x-1)\), leaving a common \((x-1)\) factor.

Key concepts

Factoring Polynomials

Factoring polynomials is a crucial step in simplifying rational expressions. A polynomial is an expression made up of variables, constants, and exponents combined using addition, subtraction, and multiplication.To factor a polynomial, you need to break it down into simpler products of smaller polynomials that, when multiplied together, will give you the original expression. This is similar to finding factors of numbers but applied to algebraic expressions.Here's how you can factor a quadratic polynomial:

• Identify the polynomial: Look for a polynomial in the form of \(ax^2 + bx + c\).
• Find two numbers: These numbers should multiply to \(ac\) (the product of the first and last coefficient) and add to \(b\) (the middle coefficient).
• Rewrite the middle term: Use the two numbers found in the step above to split the middle term.
• Factor by grouping: Group the terms into two pairs and factor out the common factor from each pair.
• Write the final expression: Combine the factors into a single expression.

Factoring is not only important for simplification but also helps in finding the roots of equations and solving algebraic problems efficiently.

Rational Expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. Just like numerical fractions, rational expressions can often be simplified by reducing them to their most basic form.In rational expressions:

• Identify the expression: It is typically in the form \(\frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials.
• Factor both the numerator and denominator: Use factoring techniques to rewrite each polynomial as a product of simpler expressions.
• Cancel common factors: Look for any common factors between the numerator and the denominator. These common factors can be divided out, simplifying the expression.
• Check your work: Always ensure that the expression is fully simplified and that no further common factors remain.

Simplifying rational expressions can make them easier to work with and to understand their behavior, especially in more complex algebraic problems.

Common Factors

Common factors play an essential role in simplifying rational expressions. A common factor is a term that is present in both the numerator and the denominator of a rational expression. To identify common factors:

• Factorize completely: Both the numerator and the denominator should be factored fully into their simplest components.
• Look for identical factors: Once factored, identify any terms that appear in both the numerator and denominator. These terms are your common factors.
• Cancel out the common factors: Removing these means you divide them from both the numerator and the denominator, which simplifies the expression.
• Review the result: Ensure that the resulting expression no longer contains any common factors and represents the simplest form of the original expression.

Finding and canceling common factors ensures that the rational expression is as straightforward as possible, which is particularly helpful for solving mathematical equations and applying further operations.