Problem 68 - Algebra 2

Question

Writing Explain how to factor \(3 x^{2}+6 x-72\) completely.

Step-by-Step Solution

Answer

The factored form of the quadratic expression \(3x^2 + 6x - 72\) is \(3(x + 6)(x - 4)\).

Step 1: Factor out the greatest common factor

Identify the greatest common factor (GCF) of the coefficients of the terms in the polynomial. Here, the GCF of 3, 6, and -72 is 3. Divide each term by the GCF to factor it out. Thus, the expression becomes: \(3(x^2 + 2x - 24)\).

Step 2: Factor the quadratic expression

Factor the trinomial inside the parentheses: \(x^2 + 2x - 24\). We are looking for two numbers that multiply to \(-24\) (the constant term) and add up to \(+2\) (the coefficient of the \(x\) term). The numbers that work are \(+6\) and \(-4\). Rewrite the trinomial as \(x^2 + 6x - 4x - 24\).

Step 3: Use the grouping method

Group the terms of the expression into two pairs: \((x^2 + 6x)\) and \((-4x - 24)\). Factor out the common factor of each pair to get: \(x(x + 6) - 4(x + 6)\).

Step 4: Factor by grouping

Notice that \((x + 6)\) is a common factor in both terms. Factor \((x + 6)\) out of the grouped terms to get the final factored form: \(3(x + 6)(x - 4)\).

Key Concepts

Greatest Common Factor

Understanding the concept of the Greatest Common Factor (GCF) is paramount when it comes to factoring polynomials. The GCF is the largest number that divides into all the coefficients of the terms without leaving a remainder. In our example, the polynomial to be factored is 3x^2 + 6x - 72. We identify that 3 is the GCF as it is divisible into each coefficient: 3, 6, and -72. Factoring out the GCF simplifies the polynomial, making it easier to work with. The expression thus becomes 3(x^2 + 2x - 24). Not only does this step reduce complexity, but it also lays the groundwork for further factoring methods to be applied.

By extracting the GCF, students can concentrate on factoring the remaining polynomial, which is often easier to handle due to its reduced coefficients. This process encourages a smoother transition to completing the square, searching for binomial pairs, or deploying the grouping method for higher-degree polynomials. It is also the first check one should make when beginning to factor any polynomial expression.

Quadratic Expressions

Factoring quadratic expressions is a fundamental skill in algebra. A quadratic expression is generally in the form ax^2 + bx + c, where a, b, and c are constants. To factor such an expression, we seek two binomials that when multiplied together give us the original quadratic.

In the example x^2 + 2x - 24, after factoring out the GCF, our goal is to find two numbers that multiply to -24 (the c term) and add to +2 (the b term). (x + 6) and (x - 4) fit the bill, since 6 * (-4) = -24 and 6 + (-4) = 2. Thus, the quadratic expression factors to (x + 6)(x - 4). Understanding how to break down quadratics into their binomial factors is essential, especially because it allows for the simplification of equations and can help with finding the roots of the quadratic, which are the values of x where the graph of the quadratic crosses the x-axis.

Grouping Method

The grouping method is a technique used to factor polynomials with four or more terms. It involves rearranging and pairing terms that have a common factor and factoring them separately. In the provided example, we first split the middle term of the quadratic x^2 + 2x - 24 into two terms, 6x and -4x, to prepare for grouping. We then have the expression x^2 + 6x - 4x - 24, which allows us to group the terms into (x^2 + 6x) and

(-4x - 24). Within each group, we then factor out the greatest common factor: x(x + 6) and -4(x + 6). After factoring, we notice the common binomial factor (x + 6), which we factor out, resulting in the final factored form of 3(x + 6)(x - 4). This method is particularly useful for polynomials that are not easily factored by other methods, such as simple trinomials or difference of squares. It provides a structured approach to managing more complex expressions, making it an invaluable tool in a student's algebra toolkit.