Problem 30 - Algebra 2 and Trigonometry

Question

In \(27-39,\) factor each polynomial completely. $$ 4 a x^{2}+4 a x-24 a $$

Step-by-Step Solution

Answer

The completely factored form is \(4a(x + 3)(x - 2)\).

Step 1: Identify Common Factors

Look at each term in the polynomial \(4ax^2 + 4ax - 24a\) and identify the greatest common factor (GCF). The GCF here is \(4a\) because each term shares this factor.

Step 2: Factor Out the GCF

Factor out the common factor \(4a\) from each term:\[4ax^2 + 4ax - 24a = 4a(x^2 + x - 6)\]

Step 3: Factor the Quadratic Expression

Focus on factoring the quadratic \(x^2 + x - 6\). Look for two numbers that multiply to \(-6\) and add to \(1\), which are \(3\) and \(-2\).

Step 4: Write the Factored Form

Express \(x^2 + x - 6\) as a product of two binomials using the numbers found:\[x^2 + x - 6 = (x + 3)(x - 2)\]

Step 5: Combine the Factors

Combine the results from steps 2 and 4 to write the completely factored form of the polynomial:\[4ax^2 + 4ax - 24a = 4a(x + 3)(x - 2)\]

Key Concepts

Greatest Common Factor

When you begin the process of factoring a polynomial, one of the first steps is to identify the greatest common factor (GCF). The GCF is the largest factor that divides each term of the polynomial without leaving a remainder. In simpler terms, it's the biggest number (or algebraic term) that you can "pull out" from all terms of the polynomial.

For instance, in the polynomial \(4ax^2 + 4ax - 24a\), each term has factors of \(4a\). Therefore, the GCF is \(4a\), and it can be factored out from the entire expression. By doing so, the expression transforms into a much simpler form:
\[4ax^2 + 4ax - 24a = 4a(x^2 + x - 6)\]

Finding a GCF not only simplifies the polynomial, making it easier to work with, but it also sets the stage for further factoring of the remaining expression. Remember, always look for the GCF first before moving on to more complex factoring steps.

Quadratic Expression

A quadratic expression is a polynomial of degree two, typically written in the form \(ax^2 + bx + c\). In our example, we've already factored out the GCF, yielding \(x^2 + x - 6\), which is a quadratic.

Factoring a quadratic expression involves finding two numbers that multiply to the constant term (\(-6\) in this case) and add to the coefficient of the middle term (\(1\)). This might seem tricky at first, but with practice, you can become adept at identifying these numbers.

Here is a step-by-step process:

• First, determine the product you need: \(-6\).
• Next, find two numbers that multiply to \(-6\) and add to \(1\). Those numbers are \(3\) and \(-2\).
  
  
• These numbers allow you to express the quadratic in its factored binomial form, which is a product of \((x + 3)\) and \((x - 2)\).

So, \(x^2 + x - 6\) is factored as \((x + 3)(x - 2)\).

Factored Form

Once you have identified the greatest common factor and dealt with the quadratic expression, the final goal is to write the entire polynomial in its factored form. This reveals the polynomial broken down into a product of simpler polynomials or terms.

After finding that \(x^2 + x - 6\) factors into \((x + 3)(x - 2)\), you can express the entire original polynomial in its fully factored form by combining all previous steps:
\[4ax^2 + 4ax - 24a = 4a(x + 3)(x - 2)\]

Why is this important? Writing a polynomial in its factored form is crucial for solving equations, simplifying expressions, and understanding the roots or solutions of a quadratic equation. In real-world applications, it allows you to quickly identify critical points and behaviors of algebraic expressions.

• It simplifies the complexity of the polynomial.
• Easy identification of polynomial roots or zeroes.
• Useful in solving algebraic fraction equations.

Understanding factored form provides a clearer, more insightful perspective of the polynomial's structure.