Problem 24 - Elementary Algebra

Question

Factor the following problems, if possible. $$ 4 x^{2}+8 x-21 $$

Step-by-Step Solution

Answer

Answer: The factored form of the given quadratic expression is \((4x-7)(x+3)\).

Step 1: Identify the common factor

First, we need to see if there is any common factor other than 1 between the coefficients 4, 8, and -21. If we look closely, we can observe that all these numbers have no common factor other than 1. Since there is no common factor, we can proceed to the next step.

Step 2: Implement factoring methods

To factor a quadratic expression, several methods can be applied. In this example, we will be using the "product-sum" method. It involves finding two numbers that multiply to the product of the first and last terms, and add up to the middle term. For this particular problem, the product will be equal to: \((4)(-21)=-84\). We want the sum to be: \(8\)

Step 3: Find the product-sum pair

Now, we must find two numbers that multiply to -84 and add up to 8. The numbers we are looking for are 12 and -7 because \((12)(-7)=-84\), and \(12+(-7)=8\).

Step 4: Substitute the middle term

Next, we will replace the middle term (8x) with these two numbers as follows: $$ 4x^2 + 12x - 7x - 21 $$

Step 5: Factor by grouping

We will now factor by grouping the terms in pairs: $$ (4x^2 + 12x) + (-7x - 21) $$ Factor out the common factors from each pair: $$ 4x(x + 3) - 7(x+3) $$

Step 6: Extract the common binomial factor

Since both terms now have a common binomial factor, \((x+3)\), we can factor it out: $$ (4x-7)(x+3) $$ This is the factored form of the given quadratic expression.

Key Concepts

Product-Sum Method

The Product-Sum Method is a popular technique for factoring quadratic expressions, specifically quadratics in the form of \(ax^2 + bx + c\). This method involves identifying two numbers. These two numbers must multiply to the product of the quadratic term \(a\) and the constant term \(c\), and simultaneously add up to the middle term coefficient \(b\). Here's how it works step by step:

• Calculate the product of the first coefficient \(a\) and the last coefficient \(c\).
• Identify two numbers that multiply to this product and add to the middle coefficient \(b\).
• Use these numbers to split the middle term \(bx\) into two separate terms.

In the case of the quadratic \(4x^2 + 8x - 21\), the product \((4)(-21) = -84\). The numbers 12 and -7 are identified because \(12 \times -7 = -84\) and \(12 + (-7) = 8\).
This gives us a clear path to rewrite the middle term and facilitate factoring using the Grouping Method.

Grouping Method

The Grouping Method is a useful process for simplifying complex expressions and finding factors, particularly after using the Product-Sum Method. Once you have split the middle term using the Product-Sum numbers, the next steps are as follows:

• Group terms into pairs.
• Factor out the greatest common factor in each group.
• Look for a common binomial factor in the grouped expression and factor it out.

Using our example \(4x^2 + 12x - 7x - 21\), we group them as \((4x^2 + 12x) + (-7x - 21)\). Factoring individual groups gives us \((4x(x + 3) - 7(x + 3))\). Notice that \((x + 3)\) is a common factor. Therefore, the expression can be factored into \((4x - 7)(x + 3)\).
This neatly simplifies the quadratic into a product of binomials, making it more manageable and ready for further algebraic manipulation if necessary.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators (such as addition and multiplication). Understanding how they can be rewritten in different ways is crucial for solving equations and performing other calculations in algebra. When factoring quadratics, expressions often need to be transformed.

• Quadratic expressions like \(ax^2 + bx + c\) can be simplified into the product of two binomials.
• This transformation is facilitated by methods like the Product-Sum and Grouping Method.
• Factoring reveals simpler components of the expression, useful for solving equations or simplifying further calculations.

Grasping how to manipulate algebraic expressions, especially recognizing patterns and using factoring techniques, is key in high-school algebra and beyond. In the example \(4x^2 + 8x - 21\), reducing it to \((4x-7)(x+3)\) not only confirms the expression's validity but also sets a strong foundation for understanding and solving more complex algebraic problems.