Problem 23 - College Algebra with Modeling and Visualization

Question

Add the polynomials. $$(4 x)+(1-4.5 x)$$

Step-by-Step Solution

Answer

The sum of the polynomials is \(1 - 0.5x\).

Step 1: Identify the like terms

In the given polynomials, identify the terms that are similar, which means they have the same variable raised to the same power. In this case, the like terms are \(4x\) and \(-4.5x\).The constant term to consider is \(1\).

Step 2: Combine the like terms

Combine the coefficients of the like terms together. The coefficients are \(4\) from \(4x\) and \(-4.5\) from \(-4.5x\). Perform the addition: \(4 + (-4.5) = -0.5\). This results in \(-0.5x\).

Step 3: Write the simplified polynomial

Combine the result from Step 2 with the constant terms. The constant term \(1\) is added to the polynomial, resulting in the final answer: \(1 - 0.5x\).

Key Concepts

Like Terms

To successfully add polynomials, understanding the concept of like terms is crucial. Like terms are terms that contain the same variable raised to the same power. For instance, in the expression \((4x) + (1 - 4.5x)\), the terms \(4x\) and \(-4.5x\) are like terms because they both include the variable \(x\) raised to the first power.
Recognizing like terms allows you to easily combine them.

• They simplify the addition process because you only need to focus on combining their coefficients.
• Ignoring unlike terms results in incorrect solutions.
• It's a vital step to ensure you don't mix them with the constant parts of the polynomial like the number \(1\) in this problem.

By keeping like terms grouped, you can proceed with updating the polynomial correctly.

Coefficients

Coefficients play an essential role in polynomial addition. A coefficient is the numerical part of a term with a variable. For the polynomial terms \(4x\) and \(-4.5x\), the coefficients are \(4\) and \(-4.5\) respectively.
When combining like terms, focus on the coefficients:

• Add or subtract the coefficients, depending on their signs. For example, \(4 + (-4.5)\) will give \(-0.5\).
• The resulting coefficient is used to form a new term with the original variable.
• If all terms cancel each other out (i.e., sum to zero), that variable term vanishes from the polynomial.

Understanding coefficients helps in simplifying the polynomial accurately, ensuring that all variables are accounted for.

Simplified Polynomial

Once the like terms are combined by adjusting their coefficients, the final stage is to write the polynomial in its simplest form. This is known as the simplified polynomial.
The goal is to ensure there are no unnecessary terms, making it easier to interpret and use in further calculations:

• Include all constant terms separately, as seen with \(1\) in the example.
• Ensure there are no more like terms that can be combined.
• The simplified polynomial for \((4x) + (1 - 4.5x)\) becomes \(1 - 0.5x\), a clean and clear expression.

Mastering the art of simplification will make complex problems more manageable and enhance computational efficiency in higher-level math.