Problem 51 - College Algebra with Modeling and Visualization

Question

Factor the expression completely. \(3 x^{3}+12 x^{2}+9 x\)

Step-by-Step Solution

Answer

The expression is factored as \(3x(x + 3)(x + 1)\).

Step 1: Identify Common Factor

Look at the terms in the expression \(3x^3 + 12x^2 + 9x\). Each term has an \(x\) and is divisible by 3. Identify the common factor, which is \(3x\).

Step 2: Factor Out the Common Factor

Factor \(3x\) out of the expression: \(3x(x^2 + 4x + 3)\). This simplifies the expression by removing \(3x\) from each term.

Step 3: Factor the Quadratic

Now focus on factoring the quadratic expression \(x^2 + 4x + 3\). We need to find two numbers that multiply to 3 (constant term) and add to 4 (linear coefficient). These numbers are 3 and 1.

Step 4: Apply the Factors

Write the expression \(x^2 + 4x + 3\) as \((x + 3)(x + 1)\). This factors the quadratic expression.

Step 5: Combine All Parts

Combine the factored parts to get the fully factored form of the original expression: \(3x(x + 3)(x + 1)\).

Key Concepts

Common Factor

When dealing with algebraic expressions like \(3x^3 + 12x^2 + 9x\), one efficient way to simplify them is by factoring out the common factor. The common factor is a term that is shared across all parts of the expression. Identifying it is the first critical step in simplifying complex expressions.

For our expression, notice each term \(3x^3, 12x^2,\) and \(9x\) has two things in common: the number 3 and the variable \(x\). This occurs because 3 is a factor of the coefficients (3, 12, and 9) and \(x\) is present in each term. By observing these common elements, you can factor them out to simplify your equation significantly.

Here's how you can apply it:

• Look for the largest number that divides all the coefficients.
• Identify any variables that appear in every term.

For our example, factoring out \(3x\) from the expression gives us \(3x(x^2 + 4x + 3)\). This step makes the equation much simpler and prepares it for further factoring.

Quadratic Expression

Once you've factored out the common term from our expression, you might notice a quadratic expression is left, which is \(x^2 + 4x + 3\). A quadratic expression is typically in the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. Here, \(a=1\), \(b=4\), and \(c=3\).

Factoring quadratic expressions involves finding two numbers that both multiply to \(c\) and add to \(b\). This method helps break down the quadratic into more manageable linear factors.

For the expression \(x^2 + 4x + 3\), look for numbers that multiply to 3 and add to 4. The numbers 3 and 1 fit these criteria perfectly.

• 3 * 1 = 3
• 3 + 1 = 4

Thus, the expression can be rewritten as \((x + 3)(x + 1)\). This step is vital as it systematically breaks down the quadratic portion into simpler terms.

Fully Factored Form

Finally, the term 'fully factored form' means that an expression has been simplified as much as possible and is expressed as a product of its factors. For the expression \(3x(x^2 + 4x + 3)\), we continue by applying the previously found factors of the quadratic portion.

By bringing together all parts, the complete factored expression is \(3x(x + 3)(x + 1)\). Each component of the expression is now a factor in its simplest form.

Breaking down the expression like this:

• \(3x\) is a factor of the common terms originally present in the equation.
• \(x + 3\) and \(x + 1\) come from the quadratic portion.

Combining all these gives you a fully factored form, meaning there are no further factors to extract. This form offers greater insights if you're solving for specific values or further algebraic operations.