91Ó°ÊÓ

In algebra, expressions represent mathematical phrases that can include numbers, variables, and operators.
The polynomials in our exercise, such as 3z^4 - 2z^3 + z - 5, are examples of algebraic expressions.
Polynomials can be defined as algebraic expressions composed of variables and constants, connected through addition, subtraction, multiplication, and non-negative integer exponents.

For example, a polynomial like 3z^4 means: 3 multiplied by the variable z raised to the power of 4.
Operations such as polynomial multiplication require a deep understanding of these expressions and their manipulation.

The distributive property is a crucial concept in algebra that simplifies multiplication over addition or subtraction.
It states that a(b + c) = ab + ac.
In polynomial multiplication, we apply the distributive property to expand the products.
For instance, in Step 2, we multiplied each term in 3z^4 - 2z^3 + z - 5 by 2z, resulting in 6z^5 - 4z^4 + 2z^2 - 10z.
The same approach in Step 3 applied the property to -5. Each term is multiplied by -5 leading to -15z^4 + 10z^3 - 5z + 25.

The distributive property helps distribute the multipliers across each term in the polynomial correctly, an essential step before combining like terms.

Combining like terms simplifies expressions by merging terms with the same variables and exponents.
In our solution, we tackled this in Step 4 and 5.
Like terms have matching variable parts. For example, -4z^4 and -15z^4 are like terms because their variable parts (z^4) are identical.
When we combined them: -4z^4 - 15z^4 = -19z^4.

This step is crucial for simplifying the final polynomial accurately.
Without combining like terms, the expression remains lengthy and harder to interpret.
As a final result, we got the simplified polynomial: 6z^5 - 19z^4 + 10z^3 + 2z^2 - 15z + 25.
Understanding how to identify and combine like terms will immensely help in handling complex algebraic expressions.