Problem 64 - Elementary and Intermediate Algebra

Question

Factor out \(-1\) from each polynomial. $$ 7-8 b $$

Step-by-Step Solution

Answer

The factored form is \\(-1(-7 + 8b)\\).

Step 1: Understand the Objective

We need to factor out \(-1\) from the given polynomial. Factoring out \(-1\) means that we will rewrite the polynomial such that \(-1\) is a factor, ultimately reversing the signs of each term within the polynomial.

Step 2: Apply Factoring

To factor out \(-1\) from the polynomial \(7 - 8b\), we change the signs of each term inside the polynomial. So: \(7\) becomes \(-7\) and \(-8b\) becomes \(+8b\).

Step 3: Rewrite the Expression

Place \(-1\) as a factor outside a parenthesis and rewrite the expression as: \(-1( -7 + 8b )\). This is the polynomial with \(-1\) factored out.

Key Concepts

Polynomials

Polynomials are algebraic expressions consisting of variables and coefficients. Each term in a polynomial can have a variable raised to an integer power. Polynomials are used extensively in mathematics to model various types of problems. They can include several terms, but each term is structured similarly featuring constants and variables combined through addition, subtraction, and multiplication. For example, a simple polynomial might look like this: \(2x^2 - 4x + 7\). Here, \(2x^2\), \(-4x\), and \(7\) are the individual terms.
Understanding the role of each component in a polynomial is crucial. Polynomials can have:

• Constants: These are standalone numbers without variables, like \(7\) in our example.
• Variables: Represent unknown values and are often denoted by letters such as \(x\) or \(b\).
• Coefficients: These are numbers that multiply variables, like \(2\) and \(-4\) in the example.

Knowing how to manipulate and factor polynomials is essential as it forms the basis for solving more complex algebraic equations.

Negative Numbers

Negative numbers are numbers less than zero and are represented with a minus sign \((-\)). Factorization often involves dealing with negative numbers, especially when simplifying or rearranging expressions.
In the context of factoring a polynomial, removing a negative sign from each term means changing the signs of all the terms involved, impacting calculations and requiring careful attention to sign changes. For example, in the process of factoring \(-1\) from the polynomial \(7 - 8b\), the signs of the coefficients in the expression reverse, switching \(7\) to \(-7\), and \(-8b\) to \(+8b\). This process exemplifies how negative numbers can change the structure of an expression through basic arithmetic operations while maintaining the equivalency of the expression.

Algebraic Expressions

An algebraic expression is a mathematical phrase combining numbers and/or variables using arithmetic operations. They can range from simple linear expressions to complex polynomials involving multiple terms.
Algebraic expressions like \(7 - 8b\) can undergo transformations such as factoring. Factoring, including factoring out negative or specific numeric factors like \(-1\), helps simplify and solve equations by revealing or isolating key components for easier manipulation.

• Components of Algebraic Expressions:
  • Terms: The parts of the expression separated by addition or subtraction, like \(7\) and \(-8b\).
  • Coefficients: The numerical part of the term, such as \(7\) and \(-8\) in the given expression.
  • Variables: Components representing unknowns like \(b\) in our example.

Through practicing factoring, students can more effectively understand how different components within algebraic expressions interact, and it aids in solving more advanced algebraic problems by breaking them down into simpler parts.