Problem 11 - Elementary Algebra

Question

Simplify each of the following expressions by using the distributive property and combining like terms. $$ a^{3}\left(a^{2}+a+5\right)+a\left(a^{4}+3 a^{2}+4\right)+1 $$

Step-by-Step Solution

Answer

Question: Simplify the expression: $$a^{3}(a^{2}+a+5)+a(a^{4}+3a^{2}+4)+1$$ Answer: The simplified expression is $$2a^{5}+a^{4}+8a^{3}+4a+1$$.

Step 1: Distributive Property

Apply the distributive property to expand the expression: $$ a^{5}+a^{4}+5a^{3}+a^{5}+3a^{3}+4a+1 $$

Step 2: Combining like terms

Now we can combine terms with the same power of a: $$ (1+1)a^{5}+(1)a^{4}+(5+3)a^{3}+(4)a+1 $$

Step 3: Simplify

Simplify the expression by adding coefficients: $$ 2a^{5}+a^{4}+8a^{3}+4a+1 $$ So, the simplified expression is: $$ 2a^{5}+a^{4}+8a^{3}+4a+1 $$

Key Concepts

Combining Like Terms

When we talk about combining like terms in algebra, we refer to the process of adding or subtracting terms that have identical variable parts. To illustrate, suppose you have a collection of apples and oranges; you can count how many apples you have and how many oranges, but you can't combine the two since they're different kinds of fruit. Similarly, in algebra, only the terms with the same variables raised to the same power can be combined.

For example, in the expression given in the exercise, we first expanded using the distributive property and then identified the like terms. Terms like a⁵ and a³ are like terms because each contains the same base, a, raised to the same exponent.
\[\text{Combining like terms: } a^5 + a^5 \text{ and } 5a^3 + 3a^3\]
After combining them, we get the terms 2a⁵ and 8a³. This step is crucial for simplifying algebraic expressions and lays a foundation for solving more complex equations efficiently.

Algebraic Expressions Simplification

The goal behind simplification of algebraic expressions is to rewrite the expression in its most concise and understandable form without changing its value. This process often involves several steps, which typically include expanding expressions using the distributive property, combining like terms, and reducing fractions, if any, to their simplest form. In the exercise provided, we simplified an algebraic expression by first expanding and then combining like terms to reduce it to fewer terms that clearly show how different powers of a contribute to the expression.

The final simplified form in the exercise is more straightforward and denotes the same quantity as the original expression. Simplification makes expressions easier to work with, especially when solving equations, graphing functions, or deriving further algebraic results.
\[\text{Final simplified form: } 2a^5 + a^4 + 8a^3 + 4a + 1\]
Understanding simplification is incredibly beneficial as it's a foundational skill used in various areas of mathematics and practical applications.

Algebraic Operations

The term algebraic operations encompasses the various procedures we use to manipulate algebraic expressions, such as addition, subtraction, multiplication, division, and exponentiation with variables. These operations follow the same basic rules that govern arithmetic on numbers, but with the addition of variables, we also need to pay attention to proper handling of the variable parts.

When we solve the given exercise, we first apply the distributive property, which is an algebraic operation that allows us to multiply a single term by each term inside a set of parentheses. To break it down:
\[\text{Distributive property: } a^3 (a^2 + a + 5) \text{ becomes } a^5 + a^4 + 5a^3\]
Then, we combine like terms, which is another crucial algebraic operation for simplification. Finally, we conclude with addition, simply adding the coefficients of like terms. Each step uses a fundamental operation of algebra, showcasing how algebraic operations are integral to simplifying expressions and solving for unknowns.