Problem 27 - Intermediate Algebra

Question

Multiply, and then simplify, if possible. See Example 3. $$ \frac{2 x^{2}-x-3}{x^{2}-1} \cdot \frac{x^{2}+x-2}{2 x^{2}+x-6} $$

Step-by-Step Solution

Answer

The simplified product is \(\frac{x + 2}{(x + 1)(x - 2)}\).

Step 1: Factor all expressions

First, we need to factorize the expressions in the numerators and denominators. - For \(2x^2 - x - 3\): it factors into \((2x + 3)(x - 1)\).- For \(x^2 - 1\): it is a difference of squares and factors into \((x + 1)(x - 1)\).- For \(x^2 + x - 2\): it factors into \((x + 2)(x - 1)\).- For \(2x^2 + x - 6\): it factors into \((2x + 3)(x - 2)\).

Step 2: Rewrite the expressions

Rewrite the original multiplication problem with the factorized forms of the polynomials: \[\frac{(2x + 3)(x - 1)}{(x + 1)(x - 1)} \cdot \frac{(x + 2)(x - 1)}{(2x + 3)(x - 2)}\]

Step 3: Cancel out common factors

Now, cancel out any common factors present in both the numerators and the denominators across the fractions. - \((x - 1)\) appears in all parts, so it can be canceled out from the numerators and denominators.- \((2x + 3)\) is present in a numerator and a denominator, so it can also be canceled out. This simplifies to: \[\frac{(x + 2)}{(x + 1)(x - 2)}\]

Step 4: Simplify the final expression

Since all cancellations have been made, the expression is already simplified. The final simplified product is: \[\frac{x + 2}{(x + 1)(x - 2)}\].

Key Concepts

Factoring Polynomials

When dealing with polynomial expressions, one foundational skill is factoring. This process involves rewriting a polynomial as a product of its factors, which are simpler polynomials. Consider the polynomial \( 2x^2 - x - 3 \). Factoring this requires finding two binomials that multiply back to the original polynomial. This particular polynomial can be expressed as \((2x + 3)(x - 1)\). It's done by identifying the coefficients that can combine and multiply to give the original term's coefficients and constant.

Another common situation is factoring special forms such as the difference of squares. An example is \(x^2 - 1\), which factors into \((x + 1)(x - 1)\). Recognizing these patterns speeds up the factoring process.

• Identify quadratic expressions that can be easily factored by inspection or using the quadratic formula.
• Look for differences of squares, which can be directly factored into conjugate binomials.
• Use common factor techniques, if possible, to simplify before applying other methods.

Understanding these techniques is crucial to simplify complex polynomial expressions, especially when multiplying or dividing them in rational expressions.

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are both polynomials. The expression given, \( \frac{2x^2-x-3}{x^2-1} \cdot \frac{x^2+x-2}{2x^2+x-6} \), is a form of rational expression. These require careful manipulation, especially multiplication or division, after factoring the polynomials involved.

To handle rational expressions efficiently, one should:

• Factor each polynomial in the numerator and the denominator separately.
• Rewrite the expression in terms of these factors, making cancellation easier.
• Cancel out any common factors present in both, which simplifies the expression, reducing complexity and potential for error.

For example, in the given step-by-step solution, common factors such as \((x - 1)\) and \((2x + 3)\) were identified and canceled out in both fractions. Finding these cancellations is key to simplifying rational expressions efficiently.

Simplifying Algebraic Expressions

Simplifying algebraic expressions, especially rational ones, involves reducing them to their simplest form. Using the given example, after factoring and rewriting the initial expression, common factors were canceled. This essential step ensures that the expression remains as straightforward as possible.

After performing all necessary cancellations, the expression becomes \( \frac{x + 2}{(x + 1)(x - 2)} \). Notice how simplification reduces potential computational errors, making further manipulation straightforward. It often helps in understanding the behavior of the expression for different values of \(x\).

• Always factor first to reveal possible simplifications.
• Perform careful cancellations, checking that terms are indeed identical across both numerators and denominators.
• Avoid dividing by zero; ensure values that make any denominator zero are outside the expression's domain.

Simplifying expressions isn't just about making them aesthetically pleasing; it can lead to better insight into their properties and solutions.