91Ó°ÊÓ

Monomials are algebraic expressions composed of a single term that includes a constant, a variable, or a combination of both.
For example, in the exercise \(-7 p^{4}\) and \2 p^{4}\ are monomials. Each has a coefficient (a numerical part) and a variable part raised to an exponent.
Here, \-7\ and \2\ are coefficients, while \p^{4}\ represents the variable part.
Key features of monomials include:

• A constant coefficient, which is any real number, like -7 or 2
• One or more variables raised to non-negative integer exponents

In summary, a monomial is a single algebraic term, and each part of it plays a crucial role in algebraic operations.

Subtraction of monomials involves finding the difference between two similar algebraic terms. In this exercise, the key steps involve:

• Identifying the monomials
• Setting up the subtraction
• Combining the like terms

For example, when asked to subtract \2 p^{4}\ from \-7 p^{4}\:
First, identify both monomials: \(-7 p^{4}\) and \(2 p^{4}\).
Next, set up the subtraction expression:
\-7 p^{4} - 2 p^{4}\.
Remember, the variable parts must be identical to perform the operation.
Here, we subtract the coefficients while keeping the variable part the same:
The calculation steps are:

• Subtract the coefficients: \-7 - 2 = -9\
• Keep the variable part untouched: \p^{4}\

So, \-7 p^{4} - 2 p^{4} = -9 p^{4}\. This straightforward process focuses on understanding like terms and performing basic arithmetic operations.

Combining like terms is a fundamental concept in algebra that simplifies expressions and solves equations. Like terms are terms that have the same variable and exponent.
For instance, in our exercise, \(-7 p^{4}\) and \(2 p^{4}\) are like terms. Both terms share the same variable, \p\, raised to the power of 4.
This common variable part allows us to combine them using arithmetic operations on their coefficients. Here’s how:

• Ensure the terms have the same variable part (both variable and exponent)
• Perform the specified operation (addition or subtraction) on the coefficients
• Keep the variable part unchanged in the result

In our example, subtracting, the variable part \p^{4}\ remains the same throughout.
By subtracting the coefficients directly, \-7 - 2 = -9\, we get \(-9 p^{4}\).
This process of combining like terms simplifies algebraic expressions and reduces complexity, making problem-solving more manageable.