Problem 70 - Algebra A Combined Function

Question

A piece of quarter-round molding is \((13 x-7)\) inches long. If a piece \((2 x+2)\) inches long is removed, express the length of the remaining piece of molding as a polynomial in \(x\).

Step-by-Step Solution

Answer

The remaining length of the molding is \(11x - 9\) inches.

Step 1: Understand the Problem

We need to find the remaining length of a piece of quarter-round molding initially measuring \((13x-7)\) inches after cutting off a piece \((2x+2)\) inches long. This involves subtracting the length of the removed piece from the original length.

Step 2: Set Up the Subtraction

We express the problem as a subtraction of polynomials. The original length of the molding is \((13x-7)\) and the length of the removed piece is \((2x+2)\). Our goal is to calculate \((13x-7) - (2x+2)\).

Step 3: Distribute Negative Sign

Apply the distributive property: \((13x-7) - (2x+2) = 13x - 7 - 2x - 2\). By distributing the negative sign across \((2x+2)\), we effectively negate each term inside the parentheses.

Step 4: Combine Like Terms

Combine the \(x\) terms and the constant terms separately: \((13x - 2x)\) and \((-7 - 2)\). This gives us: \(13x - 2x = 11x\) and \(-7 - 2 = -9\).

Step 5: Express the Final Polynomial

The result of combining like terms is \(11x - 9\). Hence, the length of the remaining piece of molding expressed as a polynomial in \(x\) is \(11x - 9\).

Key Concepts

Combining Like Terms

When working with polynomials, an essential step is combining like terms. Like terms in polynomials are terms that have the same variable raised to the same power. For instance, in the expression \(13x - 2x\), both terms are considered like terms because they both contain the variable \(x\) raised to the same power, which is 1. You can think of this as adding or subtracting apples from apples.

To combine like terms:

• First, identify terms that have the same variable and exponent.
• Then, perform basic arithmetic operations; here, we either add or subtract the coefficients of these terms.

In our exercise, combining \(13x\) and \(-2x\) results in \(11x\) because \(13 - 2 = 11\). This simplification helps in expressing a polynomial in its simplest form, making it easier to understand and work with.

Distributive Property

The distributive property is a useful technique when handling polynomial subtraction or addition, particularly when parentheses are involved. In simple terms, the distributive property allows us to "distribute" multiplication over addition or subtraction within an expression. This means when you see an expression like \(-(2x + 2)\), you need to multiply the negative through each term inside the parentheses.

Here's how you apply it:

• Take the multiplier – in this case, \(-1\) – and multiply it with each term inside the parenthesis individually.
• This results in \(-2x - 2\), where both terms have been changed in their signs.

This step is crucial because it ensures that each term is accurately adjusted when we perform the subtraction. Forgetting to apply the distributive property, especially for negative signs, can lead to errors in simplification and results. It provides a structured way to ensure every piece of the expression is correctly managed before moving on to combine like terms.

Polynomial Expressions

Polynomials are expressions that consist of variables and coefficients, involving operations of addition, subtraction, and non-negative integer exponents on the variables. They are foundational in algebra and serve as a building block for many complex equations.

A polynomial can have:

• Variables, like \(x\) or \(y\).
• Coefficients, which are numbers that multiply the variables (e.g., 13 in \(13x\)).
• Constant terms, which are standalone numbers without variables.
• Exponents, which indicate the power to which a variable is raised.

In our given problem, the expression \(13x - 7\) is a polynomial of one variable \(x\) with a degree of one. When performing operations like subtraction between polynomials, as shown in our solution, the aim is to express the result in a simplified polynomial form, which was \(11x - 9\) in this case. Understanding polynomial expressions and their components is crucial, as they form the basis for much of algebraic problem-solving.