Problem 51 - Beginning and Intermediate Algebra

Question

Factor out the greatest common factor. Be sure to check your answer. Factor out \(-1\) from \(-b+8\)

Step-by-Step Solution

Answer

Factoring out \(-1\) from \(-b + 8\) gives us \(-1(b-8)\) as the simplified expression.

Step 1: Determine the GCF

The GCF of the coefficients -1 and 8 is 1. However, we want to factor out a \(-1\) to simplify the expression.

Step 2: Factor out the GCF

Now, factor out \(-1\) from the given expression \(-b + 8\): \(-1(-b+8) = -1(-b) + (-1)(8)\)

Step 3: Simplify the expression

Multiply the \(-1\) inside the parenthesis: \(-1(-b) + (-1)(8) = b -8\)

Step 4: Verify the result

Check if the result obtained is correct by distributing the \(-1\) back to each term: \(-1(b-8) = -1 \cdot b + (-1) \cdot (-8) = -b + 8\) The factored expression \(-1 (b - 8)\) is equivalent to the original expression \(-b + 8\), so the answer is correct.

Key Concepts

Greatest Common Factor

The greatest common factor (GCF) is an essential concept in factoring expressions. It refers to the biggest number that divides two or more numbers without leaving a remainder. In algebra, finding the GCF helps in simplifying expressions by bringing out common terms.
For expressions like \(-b+8\), the GCF is not immediately obvious since the expression has both terms involving coefficients. The coefficient for \(-b\) is -1, and for 8, it's positive 8. Despite this, when working with algebraic expressions, it can be useful to think about factoring out negatives for simplification.
This is because when you decide to factor out a \(-1\), it changes the signs of the terms inside the parentheses, leading to a simplified yet equivalent expression.

Negative Coefficient

Working with negative coefficients is a common step in algebra, especially when factoring expressions. A coefficient is a number multiplied by the variable in an expression, and if it's negative, it can change how we approach factoring.
For example, in the expression \(-b+8\), \(-b\) has a negative coefficient of \(-1\). When you factor out \(-1\), this negative coefficient helps to simplify the terms by switching their signs.
This process involves:

• Multiplying the entire expression inside parentheses by \(-1\).
• The negative sign causes each term's sign to flip, resulting in \(b-8\).

Understanding this concept can make managing expressions with negatives much easier, simplifying your algebraic work.

Algebraic Simplification

Algebraic simplification is all about making expressions easier to work with or understand, and it often involves factoring. When you factor an expression, you break it down into components that, when multiplied together, give you the original expression.
Consider the expression \(-b+8\). By factoring out \(-1\), you simplify this expression to \(b-8\). This transformation is not just a neat algebraic trick—it serves a real purpose in solving equations and understanding functions more easily.
The final step in algebraic simplification often includes checking your work by distributing or multiplying through to ensure you return to the original expression.

• Start with the factored form, \(-1(b-8)\).
• Distribute \(-1\) to confirm it results in \(-b + 8\).

Finishing with this verification ensures your simplification maintains the integrity of the original expression.