Problem 59 - College Algebra

Question

In Exercises \(57-84\), factor completely, or state that the polynomial is prime. $$4 x^{2}-4 x-24$$

Step-by-Step Solution

Answer

The factorized form of the polynomial \(4x^{2}-4x-24\) is \(4(x-3)(x+2)\).

Step 1: Group the terms

Group the terms in twos. Here, the equation \(4x^{2}-4x-24\) can be rewritten as \(4x^{2}-4x - 24\).

Step 2: Factor out the Greatest Common Factor (GCF)

In each group, identify the GCF and factor it out. So, here, the GCF of \(4x^{2}\), \(-4x\), and \(-24\) is \(4\), and factoring that out gives \(4(x^{2}-x-6)\).

Step 3: Factorize the quadratic

Factorize the remaining quadratic \(x^{2}-x-6\) as \((x-3)(x+2)\). This can be achieved by finding two numbers that multiply to \(a*c\) (which is -6 here) and add up to \(b\) (which is -1 here). The numbers that meet this criterion are 3 and -2.

Step 4: Write the final expression

Combine all the factors obtained to give the final factorized expression. Hence, \(4x^{2}-4x-24\) factorizes to \(4(x-3)(x+2)\).

Key Concepts

Greatest Common Factor

The concept of the Greatest Common Factor (GCF) is fundamental in factoring polynomials. Imagine that we are looking for the largest number that can be divided evenly into all terms of a polynomial. This number is the GCF, and using it simplifies the polynomial into a factored expression. For example, in the polynomial expression \(4x^{2} - 4x - 24\), each term has a common factor of 4. This can be thought of like this:

• \(4x^{2}\) is divided by 4, giving \(x^{2}\).
• \(-4x\) is divided by 4, giving \(-x\).
• \(-24\) is divided by 4, giving \(-6\).

Thus, factoring out the GCF of 4 from all terms leaves us with \(4(x^{2} - x - 6)\). This step is crucial as it simplifies the polynomial, making it easier to work with and further factorize if necessary.

Quadratic Factorization

Quadratic factorization refers to expressing a quadratic expression, which is a polynomial of the form \(ax^{2} + bx + c\), as a product of two binomials. Once we've factored out the GCF, the next step is to work with the quadratic part. Given \(x^{2} - x - 6\), the goal is to find two numbers that multiply to \(a \,*\, c = -6\) and add up to \(b = -1\). Think of these numbers as pieces of a puzzle:

• They need to combine to fit (add to -1).
• And multiply to fill a set space (multiply to -6).

Here, the numbers are 2 and -3, since \(2 \times -3 = -6\) and \(2 + (-3) = -1\). So, \(x^{2} - x - 6\) can be factorized into \((x - 3)(x + 2)\). This reveals the structure of the quadratic, breaking it down into a more digestible form.

Polynomial Factorization

Polynomial factorization is the process of breaking a polynomial into a product of its simpler parts. Each part is a factor of the polynomial. Starting with the complete polynomial, such as \(4x^{2} - 4x - 24\), the goal of factorization is to express it as a product of lower-degree polynomials.Here’s how it is done step by step:

• First, identify and factor out the GCF, as discussed earlier.
• Next, focus on the quadratic expression, \(x^{2} - x - 6\), by finding patterns or numbers that fit multiplication and addition rules (quadratic factorization).
• After determining the numbers that work for the quadratic (like -3 and 2), use them to write the quadratic as a product of binomials \((x - 3)(x + 2)\).

Combine the results: The original polynomial \(4x^{2} - 4x - 24\) becomes \(4(x-3)(x+2)\). This final expression is completely factorized, showing clearly the components that form the original polynomial. This not only makes numerical solutions easier to find but also provides insights into the behavior and properties of the polynomial.