Problem 8 - Elementary and Intermediate Algebra

Question

Fill in the blanks. The statement \(x^{2}-x-12=(x-4)(x+3)\) shows that \(x^{2}-x-12\) factors into the ____ of two binomials.

Step-by-Step Solution

Answer

Product

Step 1: Identify the Polynomial

We begin by identifying the given polynomial expression on the left side of the equation: \(x^2 - x - 12\). This is a quadratic polynomial, which means it can potentially be factored into a product of two binomials.

Step 2: Recognize the Factoring Form

The equation provided shows \(x^2 - x - 12\) rewritten as \((x - 4)(x + 3)\). This indicates that the quadratic polynomial on the left is factored into a product of two binomials: \((x - 4)\) and \((x + 3)\).

Step 3: Define the Term 'Product'

In algebra, when a polynomial is rewritten as a multiplication of polynomials, it is called a product. Hence, when \(x^2 - x - 12\) is rewritten as \((x - 4)(x + 3)\), it's being expressed as the product of these two binomials.

Key Concepts

Quadratic Polynomial

A quadratic polynomial is a type of algebraic expression that can be represented in the form of \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is a variable. These polynomials contain three terms and the highest power of the variable is two. This is why they are called 'quadratic,' stemming from the word 'quadratus,' which means "square" in Latin.

In our exercise, the polynomial is \(x^2 - x - 12\). This is a simple quadratic polynomial where \(a = 1\), \(b = -1\), and \(c = -12\). The key to working with quadratic polynomials is understanding that they can often be rewritten or "factored" into simpler expressions, making them easier to analyze and solve. Factoring is a crucial skill in algebra, especially when solving quadratic equations.

Binomials

A binomial is an algebraic expression containing exactly two terms connected by a plus (+) or a minus (-) sign. For example, \(x - 4\) and \(x + 3\) are both binomials.

• "Bi" means two, indicating two distinct terms.
• Each term can consist of numbers, variables, or both.

Understanding binomials is important because they form the building blocks for more complex polynomial expressions. In the context of factoring quadratic polynomials, binomials become especially important.

Our exercise involves recognizing that the quadratic expression can be broken down into binomials. This makes the problem easier to tackle and provides a straightforward way to express the quadratic polynomial in its factored form.

Product of Binomials

When we factor a quadratic polynomial into a product of binomials, we are essentially writing the polynomial as a multiplication of two simpler expressions. Consider the equation \((x - 4)(x + 3)\). This expression is the product of two binomials, \(x - 4\) and \(x + 3\).

Factoring a quadratic polynomial into a product of binomials helps in multiple ways:

• It provides a means to find the roots or solutions to the quadratic equation \(x^2 - x - 12 = 0\) by setting each binomial equal to zero.
• It simplifies solving processes and allows for quick identification of solutions or roots.
• It aids in graphing the polynomial as it shows the points where the graph will intersect the x-axis.

By expressing \(x^2 - x - 12\) as the product \((x - 4)(x + 3)\), we can easily solve for \(x\) and understand the behavior of the polynomial's graph.