Problem 25 - Intermediate Algebra

Question

Multiply, and then simplify, if possible. See Example 2. $$ \frac{x^{3}+3 x^{2}-3 x-9}{x} \cdot \frac{1}{x^{2}+3 x} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{x^{2} - 3}{x} \).

Step 1: Factor the Numerator of the First Fraction

The numerator of the first fraction is a polynomial: \(x^{3} + 3x^{2} - 3x - 9\). Factor by grouping. First, group the terms: \((x^{3} + 3x^{2})\) and \((-3x - 9)\). Factor out the greatest common factor from each group: \(x^{2}(x + 3) - 3(x + 3)\). Notice \((x + 3)\) is common, so you can factor it out to get: \((x + 3)(x^{2} - 3)\).

Step 2: Observe the Denominators

The denominator of the first fraction is \(x\) and the denominator of the second fraction is \(x^{2} + 3x)\), which can be factored into \(x(x + 3)\).

Step 3: Multiply the Fractions

When multiplying fractions, multiply the numerators together and the denominators together. Multiply: \((x + 3)(x^{2} - 3) \times 1 = (x + 3)(x^{2} - 3)\). The denominator becomes: \(x(x^{2} + 3x) = x^{3} + 3x^{2}\). So, the product of the fractions is: \(\frac{(x + 3)(x^{2} - 3)}{x(x^{2} + 3x)}\).

Step 4: Simplify the Expression

Look for common factors in the numerator and the denominator. The denominator can be rewritten as \(x(x + 3)\). The term \((x + 3)\) in the numerator cancels with \((x + 3)\) in the denominator. Thus, the simplified expression is \(\frac{x^{2} - 3}{x}\).

Key Concepts

Factoring Polynomials

To factor polynomials, we often start by looking for common factors in groups of terms. This is a strategic approach known as factoring by grouping. In the numerator of the first fraction, we have the polynomial \(x^3 + 3x^2 - 3x - 9\). Start by dividing it into two parts: \((x^3 + 3x^2)\) and \((-3x - 9)\).

Now, factor out the greatest common factor from each group:

• From \((x^3 + 3x^2)\), factor out \(x^2\), resulting in \(x^2(x + 3)\).
• From \((-3x - 9)\), factor out \(-3\), resulting in \(-3(x + 3)\).

Notice that \(x + 3\) is a common factor in both terms. You can take \(x + 3\) out of the expression, and therefore it becomes \((x + 3)(x^2 - 3)\).

This method is particularly useful because it simplifies complex expressions and makes it possible to identify and cancel common factors later, as shown in the subsequent steps.

Multiplying Fractions

Multiplying fractions is straight-forward in algebra. You multiply the numerators together to get a new numerator, and you multiply the denominators together to form a new denominator.

Here, we have:

• Numerators: \((x + 3)(x^2 - 3)\) and \(1\)
• Denominators: \(x\) and \(x^2 + 3x\)

When multiplying, first process the numerators: \[(x + 3)(x^2 - 3) \times 1 = (x + 3)(x^2 - 3)\] Thus, your numerator stays the same as it was when factored.

Now, adjust the denominator by multiplying \(x\) and \(x^2 + 3x\). If you factor \(x^2 + 3x\), it becomes \(x(x + 3)\).

This gives a new form: \[(x + 3)(x^2 - 3)\] Above: new numerator \[x(x + 3)\] Below: new denominator.

Multiplying fractions allows you to see any potential common factors that may appear in both parts of the fraction, which leads us to the last concept of simplifying.

Simplifying Expressions

Simplifying algebraic expressions involves reducing them to their simplest form. After multiplying fractions, you often end up with expressions that can be reduced by cancelling out common terms.

In our problem, we've obtained:\[ \frac{(x + 3)(x^2 - 3)}{x(x + 3)} \] You should notice that both the numerator and the denominator have \((x + 3)\) as a factor. You can cancel out \((x + 3)\) from both, simplifying it to:\[ \frac{x^2 - 3}{x} \]
Always remember to identify and cancel only like terms. This process helps in cutting down on unnecessary components, making your answer simpler and more comprehensible. Simplification doesn't change the value of the expression; it only provides a clearer and more compact version of the original, which is especially useful for further calculations or understanding.