Problem 30 - Elementary and Intermediate Algebra

Question

Multiply, and then simplify, if possible. \(\frac{p^{2}-p-6}{3 p-9} \cdot \frac{2 p^{2}-5 p-3}{p^{2}-3 p}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{(p+2)(2p+1)}{3p}\).

Step 1: Factor the First Numerator

The first fraction's numerator is a quadratic expression, \(p^{2} - p - 6\). To factor it, we need two numbers that multiply to \(-6\) and add to \(-1\). These numbers are \(-3\) and \(2\). Thus, the factored form is \((p - 3)(p + 2)\).

Step 2: Factor the First Denominator

The first fraction's denominator is \(3p - 9\). We factor out the greatest common factor, which is 3, giving us \(3(p - 3)\).

Step 3: Factor the Second Numerator

The second fraction's numerator is \(2p^{2} - 5p - 3\). We apply the method of factoring by grouping by finding numbers that multiply to \(-6\) (the product of \(2\) and \(-3\)), and add to \(-5\). These numbers are \(-6\) and \(1\). Rewriting the middle term gives \(2p^{2} - 6p + p - 3\), which factors to \((2p + 1)(p - 3)\).

Step 4: Factor the Second Denominator

The second fraction's denominator is \(p^{2} - 3p\). This can be factored by taking out the common factor of \(p\), giving us \(p(p - 3)\).

Step 5: Set Up the Expression to Multiply

Now combine the factored forms: \((p - 3)(p + 2)/3(p - 3)\) multiplied by \((2p + 1)(p - 3)/p(p - 3)\).

Step 6: Cancel Common Factors

Cancel out the common factors in the numerator and the denominator. The factor \((p - 3)\) appears in each fraction. Thus, cancel two occurrences of \((p - 3)\) from the numerator and denominator. The resulting expression is \((p + 2)(2p + 1)/(3p)\).

Step 7: Simplify the Expression

There are no more common factors to cancel. Therefore, the simplified form of the expression remains \((p + 2)(2p + 1)/(3p)\).

Key Concepts

Factoring Quadratics

Factoring quadratics is an essential skill in algebra that simplifies expressions and makes them easier to work with. A quadratic expression typically has the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. To factor a quadratic, we aim to write it as a product of two binomials.

For example, in the expression \(p^2 - p - 6\), we need to identify two numbers that multiply to \(-6\) (the constant term) and add to \(-1\) (the coefficient of \(p\)). These numbers are \(-3\) and \(2\). Thus, the expression factors into \((p - 3)(p + 2)\).

Understanding the process of factoring quadratics makes it easier to solve equations or simplify complex algebraic expressions. It's like solving a puzzle where you find the pieces that fit together to create a clearer picture.

Rational Expressions

Rational expressions are fractions that contain polynomials in their numerators and denominators. Simplifying these involves the same process as simplifying numeric fractions: by factoring and canceling common factors.

• For instance, consider the expression \(\frac{p^2 - p - 6}{3p - 9}\). First, we factor both the numerator and the denominator. In this case, the numerator becomes \((p - 3)(p + 2)\) and the denominator simplifies to \(3(p - 3)\).
• Once factored, cancel any common factors. Here, \(p - 3\) is present in both the numerator and denominator, allowing them to cancel out.

Understanding rational expressions and how to manipulate them is a powerful tool in algebra. It enables you to simplify complex expressions and solve equations more easily. As with numeric fractions, the key is identifying and canceling common factors.

Multiplying Fractions

Multiplying fractions is a straightforward yet crucial operation in algebra. To multiply two fractions, multiply the numerators together to get the new numerator, and the denominators to get the new denominator. Simplification often involves factoring both numerators and denominators first.

Take the following example: \(\frac{(p - 3)(p + 2)}{3(p - 3)} \cdot \frac{(2p + 1)(p - 3)}{p(p - 3)}\). Before multiplying, factor and simplify each fraction by canceling common terms, like \(p - 3\), easing the multiplication process.

This step-by-step simplification makes the multiplication of complex expressions much more manageable. The simplified product is \(\frac{(p + 2)(2p + 1)}{3p}\), showing how factoring before multiplying yields a cleaner and more digestible result.