91Ó°ÊÓ

When working with polynomials, one of the fundamental skills is combining like terms. Like terms are terms whose variables (and their exponents) are the same. In the exercise \(x^{2}+5.2 x+3.4\) and \(9.2 x^{2}-6.1 x+5\), the like terms are categorized by the degree of \(x\). To combine them, simply add or subtract their coefficients — the numerical part in front of each term.

- For \(x^2\) terms: \(1x^2 + 9.2x^2\) becomes \(10.2x^2\).
- For \(x\) terms: \(5.2x - 6.1x\) becomes \(–0.9x\).
- For constants, or terms without a variable: \(3.4 + 5\) becomes \(8.4\).

Ensure you align terms with the same degree of \(x\) vertically to avoid mistakes when combining them. This makes it easier to see which terms should be combined. This is an essential step because only like terms can be combined—it’s like adding apples to apples and oranges to oranges.

Polynomial arithmetic involves operations such as addition, subtraction, multiplication, and sometimes division on polynomials. In our current problem, the focus is on addition. When adding polynomials, we align them by like terms and perform the addition operation on the coefficients of these terms.

This is a straightforward process that often mirrors simple arithmetic, but with the additional step of ensuring you are only working with like terms. As with any other arithmetic operation, attention to detail, like keeping track of plus and minus signs and aligning terms properly, is crucial. When done correctly, polynomial addition can be as easy as adding together numbers, provided that you keep the variables and their exponents intact and unchanged.

Simplifying expressions is a way to write them in the most concise and effective manner possible. After combining like terms in the polynomial addition, we arrive at a simpler form of the expression that accurately represents the original quantity.

In the example, after adding each set of like terms, we simplify the polynomial to \(10.2x^2 - 0.9x + 8.4\). Simplification not only makes expressions easier to read but also easier to use in subsequent calculations.

Key things to ensure during simplification are: combining all like terms correctly, not changing any exponents in the process, and writing the terms in descending order of their degrees. This standardized form helps in further algebraic manipulations and in understanding the structure of the polynomial better.