Problem 57 - Elementary and Intermediate Algebra

Question

Factor out \(-1\) from each polynomial. $$ -a-b $$

Step-by-Step Solution

Answer

The expression (-a-b) is factored as -1(a+b) .

Step 1: Identify the Common Factor

Notice that both terms in the expression \(-a-b\) are negative.

Step 2: Factor out -1

To factor out \-1\, divide each term in the expression by \-1\: \(-a-b = -1(a+b)\).

Step 3: Verify the Factoring

Multiply \-1\ back in: \(-1(a+b) = -a-b\). The expression matches the original, so the factoring is correct.

Key Concepts

Polynomial Expressions

A polynomial expression is a mathematical phrase consisting of variables, coefficients, and operations such as addition and subtraction. In the example of

• -a-b,

we see two terms:

• -a and
• -b.

The polynomial does not include any exponents higher than one, which classifies it as a linear polynomial. Remember that polynomials can have a variety of forms, from simple expressions like the one above to more complex equations with multiple terms and variables.
The key to working with polynomials lies in understanding their structure:

• Terms: Individual components separated by '+' or '-' signs.
• Coefficients: Numbers in front of the variables that indicate how many times the variable is being multiplied.

Grasping these fundamentals will help when factoring or solving them. The example given requires factoring, which is a common way to simplify polynomials prior to solving or further manipulation.

Negative Coefficients

When a polynomial contains negative coefficients, it means that one or more of its terms involve subtraction. In our expression

• "-a-b"

we see that these coefficients are negative, as spelled out by the apparent negative sign preceding each term.
Handling negative coefficients is crucial because they affect both the direction and value of a polynomial. When you factor out a negative number, every term in the polynomial changes sign—this may seem small but makes a big difference in calculations.

• Example:
  • "- a": means subtracting a.
  • "- b": means subtracting b.

This also means that when you factor negative coefficients out, you're equivalent to redistributing that negative influence across the remaining expression. Always remember, factoring negative coefficients often reveals an underlying, often simpler, polynomial form.

Common Factor

A common factor in polynomials is a number that divides each term of the expressions. In our example, the term to factor out is

• "-1"

. This is because every term here is divisible by -1, providing us with a neater expression. To do this, simply divide each term by this common factor:

• "-a-b = -1(a+b),"

Here are the steps to identify and use a common factor:

• Search for a term or number that exists in every part of the polynomial.
• Factor it out, turning the rest of the expression inside a parenthesis.

Factoring by a common factor simplifies the polynomial, revealing underlying structures that might not be immediately obvious. It's like cleaning a foggy window to see the clear scene behind it, allowing further operations with much more clarity and ease. This specific instance with -1 not only simplifies but also potentially changes the outlook of problem-solving.