Problem 1 - Elementary and Intermediate Algebra

Question

Fill in the blanks. The trinomial \(x^{2}-x-12 _____ as the product of two binomials: \)(x-4)(x+3)$

Step-by-Step Solution

Answer

The blank should be filled with '='. The trinomial factors as \((x - 4)(x + 3)\).

Step 1: Understand the Problem

The problem requires us to express the trinomial \(x^{2} - x - 12\) as a product of two binomials. We need to find two numbers that multiply to -12 and add to -1.

Step 2: Identify Suitable Numbers

We need to find two numbers such that their product is -12 and their sum is -1. Testing combinations, we see that -4 and +3 multiply to -12 and add to -1.

Step 3: Write Binomials

Now that we have the numbers -4 and +3, we can express \(x^{2} - x - 12\) as a product of two binomials: \((x - 4)(x + 3)\).

Key Concepts

Polynomial Expressions

Polynomial expressions are mathematical expressions that involve a sum of powers in one or more variables multiplied by coefficients. When working with polynomials, it's crucial to understand how to manipulate and simplify them. Breaking down polynomials into components can often make complex problems easier to handle.

For example, the expression \( x^2 - x - 12 \) is a polynomial where each term is a power of \( x \). The term \( x^2 \) is the quadratic term, \( -x \) is the linear term, and \( -12 \) is the constant term. By factoring, we can express this polynomial as a product of simpler expressions, named binomials, to make solving or analyzing simpler problems and equations more manageable.

• Polynomials involve terms like constants, variables, and exponents.
• They are often rewritten by factoring to simplify mathematical operations.
• Factoring involves expressing a polynomial as the product of other polynomials.

Binomial Products

A binomial is a simple algebraic expression with just two terms. When you engage in multiplying binomials, you produce what is known as a binomial product. Understanding binomial products is essential because they populate many mathematical problems and provide insights into polynomial structure.

In the process of factoring the polynomial \( x^2 - x - 12 \), we looked for two numbers whose product equals -12 and whose sum equals -1. This helps us factor the expression into two binomials: \( (x - 4)(x + 3) \). These binomial factors can help solve other equations or verify if a solution is correct.

• Binomials are expressions with exactly two terms.
• Multiplying two binomials requires distributing each term in the first binomial by both terms in the second.
• This process can be remembered by the acronym FOIL (First, Outside, Inside, Last).

Quadratic Equations

Quadratic equations are polynomials of degree two, typically in the form \( ax^2 + bx + c = 0 \). Factoring quadratic equations into binomial products allows one to solve these equations much more easily.

In the given exercise, the quadratic \( x^2 - x - 12 \) was factored into \( (x - 4)(x + 3) \). By setting each binomial equal to zero, you find the solutions to the quadratic equation: \( x - 4 = 0 \) or \( x + 3 = 0 \). Solving these will give you the roots or solutions: \( x = 4 \) and \( x = -3 \). This process showcases how factoring helps solve equations.

• Quadratic equations can often be solved by factoring, which involves expressing the equation as a product of binomials.
• The solutions or roots of the equation can be found by solving these binomials set equal to zero.
• Understanding how to transition from quadratic polynomials to binomial products is key to solving quadratic equations.