Problem 61 - Elementary and Intermediate Algebra

Question

Factor out \(-1\) from each polynomial. $$ 5-x $$

Step-by-Step Solution

Answer

The factored expression is \(-1(5 + x)\).

Step 1: Identify the terms in the polynomial

The given polynomial is \(5 - x\). It has two terms: a constant term \(5\) and a linear term \(-x\).

Step 2: Factor out \(-1\) from the polynomial

To factor out \(-1\), change the sign of each term inside the polynomial. The constant term \(5\), when multiplied by \(-1\), becomes \(-5\), and the linear term \(-x\) becomes \(x\).

Step 3: Write the factored expression

After factoring out \(-1\), the polynomial \(5 - x\) becomes \(-1(5 + x)\).

Key Concepts

Understanding Polynomial Terms

A polynomial is made up of terms that are separated by addition or subtraction signs. Each term can vary in kind, and understanding this helps in operations like factoring. In this article, we'll delve into the types of polynomial terms you might encounter.

• **Terms**: Parts of the expression divided by + or - signs.
• Each term in a polynomial consists of a coefficient and a variable. If there is no variable, the term is constant.

An example of a polynomial is: \(3x^2 + 4x - 5\). Here, each part separated by a plus or minus sign is a term.

• **Example Terms**:
  • \(3x^2\): a quadratic term because of \(x^2\).
  • \(4x\): a linear term.
  • \(-5\): a constant term.

Breaking down polynomials into individual terms allows us to apply mathematical operations like factoring more effectively. Recognizing and categorizing terms is the first step in simplifying complex expressions.

What is a Linear Term?

In the realm of polynomials, a linear term is any term where the variable is to the first power. This means it is written as \(ax\), where \(a\) is a constant.

• **Linear Terms in a Polynomial**: They are straightforward because they have no exponents other than 1 attached to the variable.
• They may seem simple, but they play vital roles in defining the polynomial’s characteristics.

For example, in the polynomial \(5 - x\), the term \(-x\) is a linear term. It consists of the variable \(x\) and a coefficient, which can either be positive or negative. While this example has a negative sign in front of its linear term, most linear terms can also have this sign to indicate subtraction or subtraction when rearranged.

Whether we tackle complex equations or graph expressions, identifying linear terms enables us to refashion equations, factor polynomials, or shift graphs much more comfortably.

The Role of Constant Terms

A constant term in a polynomial is a term that comprises simply a number without any variables. These terms remain the same regardless of the value of \(x\).

• **Identification**: They are the numbers not multiplied by any variable.
• Even if a polynomial only has numbers, the constant may still influence calculations and outcomes.

In the expression \(5 - x\), the number \(5\) is a constant term. It's simple – no matter what value \(x\) takes, \(5\) stays as \(5\). Constant terms make equations simple and steady, acting as foundational points in many expressions.

From factoring polynomials to calculating intercepts on a graph, constant terms serve as a starting point. This stability makes them invaluable even in more complicated mathematical contexts.