Problem 53 - Algebra A Combined Function

Question

Factor \(a\) "-1" from each polynomial. $$ -x-7 $$

Step-by-Step Solution

Answer

The factored polynomial is \(-1(x + 7)\).

Step 1: Identify the Polynomial Terms

The given polynomial is \(-x - 7\). It consists of two terms: \(-x\) and \(-7\).

Step 2: Identify Common Factor

We need to factor out \(-1\) from each term in the polynomial. In both \(-x\) and \(-7\), \(-1\) can be factored out.

Step 3: Factor Out -1

When you factor out \(-1\) from \(-x\), you get \((x)\), and when you factor \(-1\) from \(-7\), you get \((+7)\).

Step 4: Write the Factored Expression

After factoring out \(-1\), the polynomial becomes: \(-1(x + 7)\).

Key Concepts

Understanding the Common Factor

In algebra, finding a common factor is a fundamental skill. It involves identifying a number or expression that divides into multiple terms evenly. When dealing with polynomials, this is crucial because it allows us to simplify expressions and solve equations more efficiently.

In the context of the exercise, the common factor to remove from the polynomial \(-x - 7\) is \(-1\). Why? Because both terms in the polynomial, \(-x\) and \(-7\), can be divided by \(-1\). This process is called 'factoring out.'

Here are a few steps to recognize a common factor:

• Identify all the terms in your polynomial.
• Look for a number or variable that is present in all terms or can divide all terms.
• Factor out the greatest number or most common expression possible across all terms.

Finding common factors is a powerful technique because it makes complex algebra operations more manageable.

Exploring Polynomial Terms

A polynomial is an expression containing multiple terms. Each term comprises a number, known as the coefficient, and a variable raised to a power. Understanding polynomial terms is essential for manipulating and simplifying these expressions.

Let's take the polynomial from the exercise, \(-x - 7\). It contains two distinct terms:

• \(-x\): This term consists of the variable \(x\) with a coefficient of \(-1\).
• \(-7\): This is a constant term containing no variable, only a coefficient of \(-7\).

These individual parts are crucial when performing operations like factoring.
By identifying and isolating each term, one can determine applicable elements such as common factors, making simplification possible.
Understanding terms allows us to deconstruct and rebuild polynomials, ensuring mastery of algebraic concepts.

Factoring Out Negative Numbers

Factoring out negative numbers in polynomials is a common technique used to simplify expressions or change the sign convention in an equation. It involves removing a negative sign from all terms within the polynomial, which can impact the way terms are handled and interpreted.

Let's break it down using the polynomial from our exercise: \(-x - 7\).
Our goal is to factor out \(-1\), which will reverse the sign of each term within the parentheses following the operation.
For instance:

• From \/(-x\/), factoring out \/(-1)\ turns it into \(+x\).
• For \/(-7)\, after the \(-1\) is factored out, it becomes \(+7\).

Thus, the polynomial transforms into: \(-1(x + 7)\). This process is valuable because it can make further computations easier or more intuitive.

Why is this important? In many mathematical contexts, a positive polynomial within parentheses is simpler to work with, especially in subsequent operations like solving equations or graphing functions. Factoring out negative numbers is hence not just a simplification, but also a strategic maneuver in working with expressions.