Problem 14 - Elementary and Intermediate Algebra

Question

Simplify each polynomial and write it in descending powers of one variable. $$ 15 x^{2}+10 x^{2} $$

Step-by-Step Solution

Answer

The simplified polynomial is \(25x^2\).

Step 1: Identify Like Terms

First, notice that both terms in the polynomial are like terms, since they both have the same variable raised to the same power: \(x^2\). The terms are \(15x^2\) and \(10x^2\).

Step 2: Combine Like Terms

Add the coefficients of the like terms together. The coefficients are 15 and 10. \(15+10=25\).

Step 3: Rewrite the Simplified Polynomial

Combine the result from Step 2 with the common variable part to form the simplified polynomial, \(25x^2\).

Key Concepts

Understanding Like Terms in Polynomials

In the world of polynomials, "like terms" are your best friends when it comes to simplification. Essentially, like terms are terms that have the exact same variable raised to the same power. For example, in the polynomial given in the exercise, both terms were like terms because they contained the variable \(x\) raised to the power of 2.

Here’s how you can identify like terms:

• Check for terms using the same variable.
• Ensure that the exponents (powers) on the variables are identical.

For instance, \(15x^2\) and \(10x^2\) are like terms because they share the variable \(x^2\). It doesn't matter what the coefficients are (that's the numbers in front of the variable). What matters is the variable and its power. Knowing how to spot like terms helps make the process of simplifying polynomials much easier.

Mastering the Art of Combining Like Terms

Once you've identified like terms, the next step is "combining" them. This simplifies the polynomial into a more concise expression. When you combine like terms, you actually just work with the coefficients—the numbers in front. For our example, this meant adding together 15 and 10.

This process involves:

• Adding or subtracting the coefficients of the like terms.
• Keeping the variable part untouched.

So, for the exercise polynomial, the steps looked like this: add 15 and 10 to make 25, and keep \(x^2\) unchanged. This gives you \(25x^2\).

This process is essential because it not only makes the expression simpler to understand but also prepares equations for further operations like solving or graphing. Remember, the key is focusing solely on the number part when combining; the variable and its power simply tag along.

Why We Use Descending Powers

Writing polynomials in "descending powers" might seem like a stylistic choice, but it plays a significant role in organization and clarity. Descending powers are when terms are ordered from highest to lowest in terms of their exponents.

The reasoning includes:

• Creating a standard format that is universally recognized.
• Making it easier to identify the degree of the polynomial, which is the highest power present.
• Simplifying the process of comparing and calculating with different polynomials.

For our sample problem, although it only had one term \(25x^2\), it's automatically in descending order, as there are no additional terms with other powers of \(x\) to reorder.

When you work with more complex polynomials, this ordering becomes invaluable. It helps streamline more complicated operations like division, factoring, or solving equations involving polynomials.