Problem 45 - Algebra 2

Question

Simplify. $$ \frac{1}{2} a^{3}\left(4 a-6 b+8 a b^{4}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \( 2a^4 - 3a^3b + 4a^4b^4 \).

Step 1: Distribute the Coefficient

Distribute the term \( \frac{1}{2} a^3 \) across each term inside the parenthesis. This means multiplying \( \frac{1}{2} a^3 \) with each term separately.- First term: \( \frac{1}{2} a^3 \times 4a = 2a^4 \)- Second term: \( \frac{1}{2} a^3 \times (-6b) = -3a^3b \)- Third term: \( \frac{1}{2} a^3 \times 8ab^4 = 4a^4b^4 \)

Step 2: Combine the Terms

Now that we have the distributed terms, combine them into a single expression:- The result from Step 1 gives us: \( 2a^4 - 3a^3b + 4a^4b^4 \)This expression is the simplified form of the original question.

Key Concepts

Distribution Property

When you encounter an expression in the form of a product, like our original exercise \[ \frac{1}{2} a^3 (4a - 6b + 8ab^4) \],it's essential to utilize the Distribution Property.This property allows us to "distribute" a term outside the parentheses across each term inside the parentheses by multiplying them.Here's a breakdown of how we applied this in our solution:

• Multiply the first term: The coefficient \( \frac{1}{2} a^3 \) is multiplied by the first term inside, \( 4a \). The result is \( 2a^4 \). Notice how the exponents for \( a \) are added: \( 3 + 1 = 4 \).
• Multiply the second term: Next, we multiply \( \frac{1}{2} a^3 \) with \( -6b \). This yields \( -3a^3b \).
• Multiply the third term: Lastly, \( \frac{1}{2} a^3 \) is multiplied by \( 8ab^4 \), resulting in \( 4a^4b^4 \). Again, be mindful of the exponents: the \( a \) terms' exponents are added here as well.

The Distribution Property helps simplify complex expressions by breaking them down into smaller, manageable parts, making it easier to perform further algebraic operations.

Combining Like Terms

After distributing the terms, you might notice some terms that can be 'combined'.Combining like terms is a process used to simplify polynomial expressions further and it involves grouping terms with the same variable parts. Remember, only terms with exactly the same variable parts and exponents can be combined together.
At the stage right after distribution in our exercise, we have the expression:\[ 2a^4 - 3a^3b + 4a^4b^4 \]Now, let's quickly analyze whether any of these are like terms:

• Term 1 and Term 3: The terms \( 2a^4 \) and \( 4a^4b^4 \) are unlike because they differ in their variable parts and exponents.
• Term 2: This term, \( -3a^3b \), stands alone as its exponent and variable parts do not match any other term in the expression.

Thus, in our solution, there are no like terms to combine further. The expression remains simplified just by distributing and assembling the terms.

Polynomial Expressions

Polynomial expressions, like the one in our exercise, are algebraic expressions that involve a sum of powers of variables with coefficients. A polynomial can have constants, variables, and exponents, entirely consisting of non-negative integers.Understanding these components is crucial in algebra as they form the basis for more complex equations and functions.
Let’s break down our simplified result from the original problem step by step:\[ 2a^4 - 3a^3b + 4a^4b^4 \]

• Classification: This expression has three separate terms, each a monomial, which collectively make up a polynomial.
• Terms:
  - The first term, \( 2a^4 \), is a basic monomial with coefficient 2 and degree 4.
  - The second term, \( -3a^3b \), has a coefficient of -3, and its overall degree is 4 (3 from \( a \) and 1 from \( b \)).
  - The third term, \( 4a^4b^4 \), incorporates both \( a \) and \( b \), with an overall degree of 8 (4 from \( a \) and 4 from \( b \)).
• Simplification: Recognizing and understanding polynomials helps in identifying ways to simplify and solve algebraic expressions efficiently.

Grasping these components equips you with the ability to solve and simplify polynomial expressions systematically, a key skill in advancing your mathematical proficiency.