Problem 19 - Algebra 2 and Trigonometry

Question

In \(9-26,\) write each expression as the product of two binomials. $$ x^{2}-x-6 $$

Step-by-Step Solution

Answer

The expression is \((x + 2)(x - 3)\).

Step 1: Understand the Problem

We are given the expression \(x^2 - x - 6\) and need to express it as the product of two binomials. This means we want to find two binomials whose product is equal to this quadratic expression.

Step 2: Identify Coefficients and Solutions

The expression is of the form \(ax^2 + bx + c\), where \(a = 1\), \(b = -1\), and \(c = -6\). We need two numbers that multiply to \(c = -6\) and add to \(b = -1\).

Step 3: Find Suitable Numbers

Consider pairs of factors of \(-6\): \((2, -3)\) and \((-2, 3)\). The pair that sums up to \(-1\) is \(2\) and \(-3\). Thus, the suitable numbers are \(2\) and \(-3\).

Step 4: Write the Factorized Form

Using the numbers \(2\) and \(-3\), we can write the factorized form as \((x + 2)(x - 3)\). Check: \((x + 2)(x - 3) = x^2 - 3x + 2x - 6 = x^2 - x - 6\), confirming the factorization is correct.

Key Concepts

Binomials

A binomial is a polynomial expression with exactly two terms. These terms are usually connected by a plus "+" or a minus "-" sign. For instance, expressions like \(x + 2\) or \(x - 3\) are classic examples of binomials.

When factoring quadratic expressions, we often end up with two binomials multiplied together. This transformation helps simplify expressions and solve quadratic equations. Here’s why binomials are important:

• They form the building blocks for more complex polynomials and enable us to express polynomials in different forms.
• Binomials are often used in factoring because multiplying two binomials is straightforward and helps simplify problems.
• Learning to manipulate them is foundational for algebra and higher-level math.

Understanding binomials is crucial, as they operate like puzzle pieces in algebra. Being able to quickly identify, manipulate, and multiply them is essential for mastering more advanced math concepts.

Quadratic Expressions

Quadratic expressions are polynomials that include a variable squared, the equation of the standard form being \(ax^2 + bx + c\). In any quadratic, \(a\), \(b\), and \(c\) represent constants, with \(a\) not being zero.

Here's why they matter:

• Quadratic equations model various real-world scenarios, like projectile motion or area calculations.
• They are pivotal for algebra, serving as a stepping stone to more advanced topics like calculus.
• Quadratic expressions are often simplified or solved by factoring, breaking them down into products of binomials.

Quadratic expressions are typically tackled using factorization when you need to find the roots or simplify the expression. This technique involves expressing the quadratic as a product of two binomials, as you’ve done with \(x^2 - x - 6\) to get \((x + 2)(x - 3)\). By learning to factorize quadratics, you gain a powerful tool for solving equations and understanding relationships between variables.

Factor Pairs

Factor pairs are pairs of numbers that, when multiplied together, give a specific product. For the quadratic expression \(x^2 - x - 6\), our goal was to find two numbers that multiply to \(-6\) (constant term) and add up to \(-1\) (linear coefficient).

Finding factor pairs involves:

• Listing all pairs of integers that multiply to the given constant term.
• Checking each pair to see if their sum equals the middle coefficient of the quadratic expression.
• Selecting the correct pair to use in factorization, as seen with the numbers \(2\) and \(-3\) for \(x^2 - x - 6\).

Mastering factor pairs helps in efficiently breaking down quadratic expressions into simpler parts. Recognizing and calculating these pairs is particularly important because it lays the foundation for solving more complex algebraic equations.