Problem 15 - Elementary and Intermediate Algebra

Question

Simplify each polynomial and write it in descending powers of one variable. $$ 18 x^{2}-19 x+2 x^{2} $$

Step-by-Step Solution

Answer

The simplified polynomial is \(20x^2 - 19x\).

Step 1: Identify Like Terms

Look at the polynomial \(18x^2 - 19x + 2x^2\). Identify the like terms, which in this case are the terms with \(x^2\) powers: \(18x^2\) and \(2x^2\).

Step 2: Combine Like Terms

Add the coefficients of the like terms together. For \(x^2\) terms: \(18 + 2 = 20\). This simplifies the polynomial to \(20x^2 - 19x\).

Step 3: Rewrite in Descending Order

Make sure the polynomial is already written in descending order of powers of \(x\). The terms are \(20x^2\) and \(-19x\), which are in descending order. So the polynomial stays \(20x^2 - 19x\).

Key Concepts

Descending Powers

When simplifying a polynomial, it’s important to order the terms by their powers. This means arranging them from the highest power to the lowest power of a chosen variable. In our exercise, the variable is \(x\).

Descending order makes it easy to read the polynomial. You'll immediately know the dominant term and how each term contributes to the polynomial's overall value. For instance, in the expression \(20x^2 - 19x\), \(x^2\) is the highest power term and comes first, followed by the \(-19x\) term.

Why is this important? Ordering by descending powers helps in comparing polynomials and is crucial for operations like polynomial addition, subtraction, and division. If the polynomial isn’t already arranged, reorder it accordingly. However, our final simplified polynomial \(20x^2 - 19x\) is correctly arranged. Always check this arrangement in any given polynomial exercise.

Like Terms

Like terms are terms in a polynomial that have identical variables raised to the same powers. Identifying and working with like terms simplifies polynomial manipulation.

In our exercise, like terms were crucial for simplifying the expression. We encountered \(18x^2\) and \(2x^2\), both terms contain \(x^2\). These are considered like terms because they share the same variable and exponent.

Recognizing like terms allows you to combine them, which simplifies computation. Think of it as adding or subtracting constants, but you're operating on coefficients of these terms. Understanding and identifying like terms is a foundational skill in algebra that aids in further algebraic manipulations.

Combining Coefficients

Combining coefficients is all about simplifying like terms by adding or subtracting their numerical coefficients. This concept is key to polynomial simplification.

In our example, we needed to add the coefficients of like terms: the \(18x^2\) and \(2x^2\) terms. Here, the coefficients \(18\) and \(2\) were added giving \(20x^2\).

Why do we do this? Combining coefficients reduces the polynomial to its simplest form. This ensures you're working with the least complicated version of the polynomial, which is not just efficient but also minimizes errors in further calculations. Always make sure to only combine terms that share the same variable and exponent to maintain the polynomial’s structure.