Problem 92 - Elementary Algebra

Question

For the following problems, perform the multiplications and combine any like terms. $$ a(a-3)(a+5) $$

Step-by-Step Solution

Answer

Question: Simplify the expression: \((a)(a-3)(a+5)\) Answer: After simplifying the expression, it becomes \(a^3 + 2a^2 - 15a\).

Step 1: Multiply the first two sets of parentheses

To multiply the first two sets of parentheses, we'll use the distributive property. Multiply the term 'a' by both terms in the second parenthesis: $$ a(a-3) = a^2 - 3a $$

Step 2: Multiply the result by the third parenthesis

Now, we have to multiply the resulting polynomial by the third parenthesis \((a+5)\). To do this, we use the distributive property again: $$ (a^2 - 3a)(a+5) $$ We'll multiply each term of the first polynomial by each term of the second polynomial: $$ a^2(a) + a^2(5) - 3a(a) - 3a(5) $$

Step 3: Simplify the resulting expression

Now, we'll simplify the expression by performing the multiplications: $$ a^3 + 5a^2 - 3a^2 - 15a $$

Step 4: Combine like terms

Finally, combine the like terms in the resulting expression: $$ a^3 + (5a^2 - 3a^2) - 15a = a^3 + 2a^2 - 15a $$ The simplified expression after performing the multiplications and combining the like terms is: $$ a^3 + 2a^2 - 15a $$

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra, making it easier to handle multiple terms within parentheses. It allows us to multiply a single term by a group of terms inside the parentheses. Here's how it works. Imagine you have an expression like \( a(b + c) \). To simplify it, you distribute the \( a \) across the terms within the parentheses:

• Multiply \( a \) by \( b \), which gives \( ab \).
• Multiply \( a \) by \( c \), which gives \( ac \).

The result is \( ab + ac \). This is exactly what we use when multiplying the terms in the original exercise like \( a(a-3) \), turning it into \( a^2 - 3a \). Always remember that the distributive property is about spreading the single term over all terms in parentheses, aiding in breaking down more complex expressions.

Combining Like Terms

Combining like terms is the next step in simplifying algebraic expressions. Once we have expanded an expression using the distributive property, it often includes similar terms we can combine.
An example from our exercise can help clarify this concept. After multiplying through the distributive property, we were left with:

• \( a^3 \)
• \( 5a^2 \)
• \(-3a^2 \)
• \(-15a \)

Our goal is to identify and combine similar terms. "Like terms" are those with the same variable raised to the same power. Here, \( 5a^2 \) and \(-3a^2 \) can be combined. Adding them gives \( 2a^2 \). The expression now simplifies to \( a^3 + 2a^2 - 15a \). Combining like terms simplifies the expression and makes it cleaner and easier to understand.

Simplifying Expressions

Simplifying expressions is about rewriting them in the simplest form, reducing complexity while maintaining the expression’s value. This involves several steps, including applying the distributive property and combining like terms. The end goal is a neat and concise expression.
In our original exercise, we moved from \( a(a-3)(a+5) \) to the final expression \( a^3 + 2a^2 - 15a \) using these techniques. By distributing and then combining like terms, we were able to transform a product of polynomials into a single polynomial expression.
Remember, simplifying makes it easier to understand and work with algebraic expressions, especially when they’re part of larger problems. It clears unnecessary clutter and focuses on what's essential: the terms and relationships that define the algebraic expression.