Adding Monomials in Algebra

6 Questions

What is the key requirement for two numbers to be considered like terms?

They must have the same base and power

Which of the following is not a common variable used in monomials?

What does an exponent indicate in an expression?

Multiplication

Which component allows us to adjust the size of terms in a polynomial without changing their structure?

Coefficients

In the expression 4x^2, what is the coefficient?

Why is it important for like terms to have the same base and power?

To simplify combining terms mathematically

Study Notes

Adding monomials is one of the fundamental operations in algebra. It involves combining two expressions with one variable by adding their coefficients while preserving their exponent. This process can help simplify more complex equations later on. Here's how it works:

Like Terms: When you combine numbers with different bases and powers, they must have the same base and power. These types of numbers are called like terms. For example, 5x^2 and 7x^2 are both like terms because they share the common factor (x^2).

Variables: Monomials usually contain variables such as x, y, z, etc., which represent unknown values. Common variables used when learning this concept are x, y, and z. However, any letter from the alphabet could theoretically serve as a variable.

Exponents: Exponentiation is another important aspect of monomials. An expression containing a number followed by a raised symbol represents multiplication of the base times itself as many times as indicated by the exponent. A monomial consists of a single term, so its exponent will always pertain to only one term.

Coefficients: Coefficients represent the multiplier before each term in a polynomial equation. They allow us to adjust the size of terms without changing their structure, so we keep them separate from the terms themselves. In other words, if there were ten apples and eight people shared them evenly, each person would get (\frac{1}{8}) of all the apples left, representing the coefficient of that portion.

When adding monomials, first make sure all terms being added are like terms; that means they consist of the same variable(s), if any, and their exponents are equal. Then, simply add corresponding coefficients together. If you encounter mixed-base exponential terms, convert those into like terms. The sum of these converted like terms gives the resultant monomial.

For instance, to add 3xy + 7xz, you check if the terms have like factors. Since neither term has like factors at a higher level, move down further until basic units start appearing. Spreading out the terms shows 3y * x + 7z * x. Now, the terms have like factors: x, and y, respectively. Therefore, the final answer becomes 4(x+y)+9x=13x+4y.