Problem 22 - Technical Mathematics with Calculus

Question

Factor completely, by hand or by calculator. Check your results. The General Quadratic Trinomial. $$7 x^{2}+23 x+6$$

Step-by-Step Solution

Answer

\(7x^2 + 23x + 6 = (7x + 2)(x + 3)\)

Step 1: Find two numbers that multiply to \(7 \times 6\) and add to 23

We are looking for two numbers that multiply to the product of the coefficient of \(x^2\) and the constant term (\(a \times c\), where \(a = 7\) and \(c = 6\)), and that add up to the coefficient of the \(x\) term (\(b\), where \(b = 23\)). So, we need numbers that multiply to \(42\) and add to \(23\).

Step 2: Find the numbers

The two numbers that meet the criteria are 2 and 21 because \(2 \times 21 = 42\) and \(2 + 21 = 23\).

Step 3: Rewrite the middle term

Use the numbers found to express the middle term \(23x\) as the sum of two terms. This gives us \(7x^2 + 2x + 21x + 6\).

Step 4: Factor by grouping

Group the terms to make it easier to factor. Group \(7x^2 + 2x\) and \(21x + 6\) to get \(x(7x+2) + 3(7x+2)\).

Step 5: Factor out the common binomial factor

Factor \(7x + 2\) out of both groups to get \(7x + 2\) as a common factor and write down the factored form as \( (7x + 2)(x + 3)\).

Key Concepts

Quadratic Equations

Quadratic equations are foundational in algebra and are identified by the highest power of the variable being squared. They take on the general form \( ax^2 + bx + c = 0 \) where \( a \) is the coefficient of the squared term, \( b \) is the coefficient of the linear term, and \( c \) is the constant. To solve these equations, one of the most common methods is factoring, where we convert the quadratic equation into a product of two binomials that equal zero.

For example, in the exercise provided, the aim is to factor the quadratic trinomial \( 7x^2 + 23x + 6 \) completely. This process allows us to find the values of \( x \) that satisfy the equation \( 7x^2 + 23x + 6 = 0 \). Factoring is particularly useful because it provides a clear, visual representation of these solutions as the points where the graph of the quadratic equation crosses the x-axis.

Factor by Grouping

Factor by grouping is a technique used to manage more complex trinomials and higher-order polynomials. This method involves reorganizing the terms and then grouping them in such a way that each group has a common factor. It becomes clearer and more straightforward to factor out these common elements, which simplifies the expression into a product of binomials or other polynomials.

During the problem-solving process from the original exercise, we arrive at the step where we must write the middle term as the sum of two terms (\( 2x + 21x \) in this case) so we can group them with other terms (\( 7x^2 \) and \( 6 \) respectively). This creates two pairs: \( 7x^2 + 2x \) and \( 21x + 6 \), making it apparent that each can be factored further. This step is pivotal, as the exercise aims to guide the student to see patterns and use them to simplify expressions to their factored form.

Binomial Factor

A binomial factor is a polynomial with two terms, typically found after factoring a larger polynomial expression. In the context of quadratic equations, the factored form often consists of two binomial factors. Once the common factor is found in the grouped terms, factoring out this binomial reveals the simpler entities that multiply to give the original trinomial.

In the textbook solution, after applying the factor by grouping method, we identify the common binomial factor \( 7x + 2 \) from both grouped terms. This binomial is then factored out, resulting in the expression being rewritten as the product of two binomials: \( 7x + 2 \) and \( x + 3 \). The final factored form, \( (7x + 2)(x + 3) \) makes it explicit that for the original quadratic equation to equal zero, one or both of these binomials must equal zero—providing the solutions to the original quadratic equation when solved for \( x \).

Having the ability to spot and factor out a binomial factor plays a crucial role in solving more complex algebraic expressions and is an essential skill for students' mathematical toolkits.