Problem 25 - Elementary and Intermediate Algebra

Question

The following is a list of random factoring problems. Factor each expression. If an expression is not factorable, write "prime." See Examples 1-5. $$ x^{2}+7 x+1 $$

Step-by-Step Solution

Answer

The expression \(x^2 + 7x + 1\) is prime.

Step 1: Identify Coefficients

Identify the coefficients from the expression \(x^2 + 7x + 1\). The expression is in the form \(ax^2 + bx + c\), where \(a = 1\), \(b = 7\), and \(c = 1\).

Step 2: Check for Factorability

To factor the quadratic expression, we need to find two numbers that multiply to \(a \times c = 1 \times 1 = 1\) and add up to \(b = 7\).

Step 3: Attempt to Find Such Numbers

List the factor pairs of 1: (1,1) and (-1,-1). Check these pairs to see if they add up to 7.

Step 4: Conclude Factorability

The pairs (1,1) and (-1,-1) do not add up to 7. Therefore, the expression \(x^2 + 7x + 1\) cannot be factored using integers.

Key Concepts

Coefficient Identification

Understanding how to identify coefficients is the first step in working with quadratic expressions. Coefficients are numerical values that multiply with variables in an expression. In the expression \(x^2 + 7x + 1\), the structure follows the standard quadratic form \(ax^2 + bx + c\). Here:

• \(a\) represents the coefficient of \(x^2\), which is 1.
• \(b\) is the coefficient of \(x\), which is 7.
• \(c\) is the constant term, which is 1.

Recognizing these coefficients is crucial because they determine the characteristics and solutions of the quadratic equation. Each variable and its corresponding coefficient play a role in the factorization process, where the goal is to express the quadratic as a product of simpler binomials.

Factorability Check

After identifying the coefficients, the next step is to check if the quadratic expression is factorable. This process involves finding two numbers that meet specific requirements:

• They should multiply together to equal the product of \(a\) and \(c\).
• They should add up to be equal to \(b\).

For the expression \(x^2 + 7x + 1\), we calculate the product \(a \times c\), which is \(1 \times 1 = 1\). Next, we seek numbers that multiply to 1 and add up to 7. By listing possible factor pairs of 1, such as (1,1) and (-1,-1), it becomes clear that neither pair sums to 7. Therefore, an integer-based factorization is not possible for this quadratic. Declaring it "prime" indicates that it cannot be factored over the integers, emphasizing the importance of thorough checks before concluding factorability.

Quadratic Expressions

Quadratic expressions are polynomials of degree two, typically written in the form \(ax^2 + bx + c\). They are fundamental in algebra due to their applications in various areas such as physics, engineering, and economics. The nature of a quadratic expression allows us to explore a wide range of problem-solving techniques, including factoring, completing the square, and using the quadratic formula.Understanding a quadratic expression involves recognizing its key components:

• The squared term \(ax^2\) defines the expression's parabola, influencing its direction and width.
• The linear term \(bx\) affects the slope and positioning of the parabola on the graph.
• The constant \(c\) shifts the entire parabola up or down.

Grasping these components provides a deeper insight into both the graphical representation and algebraic manipulation of quadratic equations. It also aids in understanding why some quadratics, like \(x^2 + 7x + 1\), may not factorize neatly over integers.