Problem 58 - Algebra A Combined Function

Question

Factor \(a\) "-1" from each polynomial. $$ 2 y-z-11 $$

Step-by-Step Solution

Answer

The factored form is \(-1(2y - z - 11)\).

Step 1: Identify the Problem

We are asked to factor out \(-1\) from the polynomial \(2y - z - 11\). This means we need to rewrite the polynomial as a product where one factor is \(-1\).

Step 2: Rewrite Each Term

Apply the factor \(-1\) to each term of the polynomial. This involves changing the sign of each term in the polynomial. Currently, the polynomial is \(2y - z - 11\). To factor out \(-1\), change \(2y\) to \(-2y\), \(-z\) to \(z\), and \(-11\) to \(11\).

Step 3: Expression with Factored Out \(-1\)

Combine the modified terms from the previous step and place \(-1\) as a factor outside a bracket: \(-1(2y - z - 11) = -(2y - z - 11) = -1(2y - z - 11)\).

Key Concepts

Factoring

Factoring in algebra means splitting an expression into multiple parts or factors that, when multiplied together, give back the original expression. It is a way to simplify expressions or solve equations.When we say we are factoring out \(-1\) from a polynomial, we essentially look for common factors that we can "take out" of the expression. Here, factoring \(-1\) involves changing the sign of each term and then writing the expression as a product where one factor is \(-1\).

• For instance, the polynomial expression \(2y - z - 11\) needs to be rewritten so that \(-1\) is outside the expression.
• This leads to changing \(2y\) to \(-2y\), \(-z\) to \(z\), and \(-11\) to \(11\).
• The factored expression is written as \(-1(2y - z - 11)\).

This process can help simplify solving further algebraic equations or expressions by reducing complexity.

Negative numbers

Negative numbers are values less than zero, and they are an important part of algebraic operations. Understanding how to manipulate negative numbers is essential, especially when factoring expressions.When factoring out a negative number, like \(-1\), you'll need to flip the signs of each term within the expression or polynomial. This flipping changes the expression so it can easily be rewritten with \(-1\) as a factor.

• In our example, the polynomial \(2y - z - 11\) becomes \(-2y + z + 11\).
• Notice each addition or subtraction sign is reversed. This ensures that when factoring \(-1\), the outcome remains mathematically equivalent to the original expression.

Grasping this concept is crucial for solving different algebraic expressions accurately, as negative numbers frequently appear and affect the terms directly.

Algebraic expressions

An algebraic expression consists of numbers, variables, and operational symbols, such as addition, subtraction, multiplication, and division. Understanding how to manipulate these expressions is key to solving algebraic problems.Polynomials are a specific type of algebraic expression that consist of variables raised to whole number powers and coefficients. Polynomials can be simple, involving only one variable, or complex, having multiple variables and terms.

• For example, in the expression \(2y - z - 11\), we see three terms: \(2y\), \(-z\), and \(-11\).
• Each term includes a coefficient and, potentially, variables raised to a power.
• Manipulating these expressions often involves operations such as factoring or distributing, where applying properties like the distributive law are essential.

Mastering algebraic expressions allows students to simplify equations and find solutions more efficiently. Understanding how terms interact through addition, subtraction, and multiplication ensures successful problem-solving in algebra.