Problem 38 - Algebra A Combined Function

Question

Factor each trinomial completely. Some of these trinomials contain a greatest common factor (other than 1). Don't forget to factor out the GCF first. $$ 4 x^{2}-4 x-48 $$

Step-by-Step Solution

Answer

The factored form of the trinomial is \(4(x - 4)(x + 3)\).

Step 1: Identify the Greatest Common Factor (GCF)

First, identify the Greatest Common Factor (GCF) for the terms in the trinomial. The terms are 4, -4, and -48. The GCF of these numbers is 4.

Step 2: Factor out the GCF

Factor out the GCF (4) from each term in the trinomial, resulting in: \[ 4(x^2 - x - 12) \]

Step 3: Set up the Factoring of the Quadratic

Now focus on the quadratic expression inside the parentheses: \(x^2 - x - 12\). Look for two numbers that multiply to -12 (the constant term) and add to -1 (the coefficient of the linear term).

Step 4: Find the Correct Pair of Factors

The pair of numbers that multiply to -12 and add to -1 is -4 and 3. Therefore, rewrite the quadratic as: \((x - 4)(x + 3)\).

Step 5: Multiply to Verify

Verify the factored expression by expanding it. Multiply \((x - 4)(x + 3)\) to ensure it simplifies back to \(x^2 - x - 12\). This confirms the factorization is correct.

Step 6: Combine the GCF with the Factored Quadratic

Finally, multiply the GCF (4) back with the factored quadratic to get the complete factorization of the original trinomial: \[4(x - 4)(x + 3)\]

Key Concepts

Greatest Common Factor

The Greatest Common Factor (GCF) is essential when simplifying polynomial expressions, including trinomials. It is the largest number by which all terms of the trinomial can be evenly divided. Identifying this factor simplifies the factorization process.

To find the GCF in a trinomial, look at the numerical coefficients and constants in each term separately. For the trinomial \(4x^2 - 4x - 48\), the coefficients are 4, -4, and -48. Breaking down these numbers, we can see that 4 is a divisor of each:

• 4: 4 divides evenly into 4
• -4: 4 divides evenly into -4
• -48: 4 divides evenly into -48

By pulling 4 out as a factor, we simplify the expression to a more manageable quadratic form. This step clears any shared multiplicatives, making it easier to handle the remaining polynomial. Understanding the GCF's role in factorization is foundational, setting the stage for more complex algebraic manipulations.

Quadratic Expressions

Quadratic expressions are polynomials that contain a variable squared \((x^2)\). They typically take the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. In the case where we factor trinomials, it's common to try and express the quadratic as a product of two binomials.

For \(4x^2 - 4x - 48\), once the GCF is removed, we are left with \(x^2 - x - 12\). This is a standard quadratic equation. The goal in factoring this expression is to find two numbers that not only multiply to \(-12\) (the constant term) but also sum to \(-1\) (the coefficient of \(x\)).

When you encounter a quadratic equation, here's what to do:

• Identify the constant (here, \(-12\)) and the coefficient of \(x\) (here, \(-1\)).
• Determine two numbers that both multiply to the constant and add to the linear coefficient.
• Express these as factors by rewriting the original quadratic expression into its binomial components.

Mastering how to work with quadratic expressions opens the door to solving various algebraic equations and is a key skill in mathematical problem-solving.

Polynomial Factorization

Polynomial factorization is an essential mathematical process often used to simplify expressions or solve equations. When dealing with trinomials like \(4x^2 - 4x - 48\), factorization helps break down the polynomial into simpler components. This involves expressing the polynomial as a product of several factors.

After identifying and factoring out the GCF from \(4x^2 - 4x - 48\), we focus on \(x^2 - x - 12\). We seek to express this quadratic polynomial as two binomials: \((x - 4)(x + 3)\). To verify the correctness of factorization:

• Multiply the binomials back using the distributive property (FOIL method).
• Ensure the expanded result equals the original quadratic \(x^2 - x - 12\).

Once confirmed, reintroduce the GCF to obtain the complete factorized form: \(4(x - 4)(x + 3)\).

Developing a strong understanding of polynomial factorization allows students to manipulate algebraic expressions more efficiently and accurately, providing a framework for solving more advanced problems across mathematics.