Problem 27 - Beginning and Intermediate Algebra

Question

Perform the indicated operations and simplify. $$(q+3)\left(5 q^{2}-15 q+9\right)$$

Step-by-Step Solution

Answer

The simplified expression after performing the indicated operations is: \( 5q^3 - 36q + 27 \).

Step 1: Distribute q

First, distribute the term q across the expression inside the second parentheses: \( q\left(5q^2 \right) \) \(q \left(-15q \right) \) \(q \left( 9 \right) \) After performing the operations, we will get: \( 5q^3 - 15q^2 + 9q \)

Step 2: Distribute 3

Next, distribute the term 3 across the expression inside the second parentheses: \( 3 \left(5q^2 \right) \) \(3 \left(-15q \right) \) \(3 \left( 9 \right) \) After performing the operations, we will get: \( 15q^2 - 45q + 27 \)

Step 3: Combine like terms

Now that we have the products from the previous steps, combine the terms by adding them together: \( 5q^3 - 15q^2 + 9q + 15q^2 - 45q + 27 \) Collapse and combine like terms: \( 5q^3 + 9q - 45q + 27 \)

Step 4: Simplify

Lastly, simplify the expression by combining the remaining like terms: \( 5q^3 - 36q + 27 \) The simplified expression after performing the indicated operations is: \( 5q^3 - 36q + 27 \)

Key Concepts

Distribution in Algebra

Distribution is a crucial concept in algebra that allows us to break down expressions into simpler parts. In its essence, it is used to eliminate the parentheses by distributing each term outside the parentheses to every term inside the parentheses.

For example, when facing an expression like

• \( (q+3)(5q^2-15q+9) \)

, we would distribute each term from

• \( (q+3) \)

to each term inside the parentheses containing

• \( 5q^2-15q+9 \)

.
This procedure involves two main steps of separate distributions:

• The term \(q\) is multiplied by each term in the second parentheses.
• Then, the term \(3\) is multiplied similarly to each term inside \(5q^2-15q+9\).

Through this process, each term in the parentheses interacts with every outside term, ensuring no value is overlooked. When done carefully, distribution makes multiplying polynomials straightforward and manageable.

Simplifying Expressions

Simplifying expressions is about making an algebraic expression more compact and easier to manage by reducing complexity without changing its value. After distribution, you will often have an expression with many terms that need reducing.

• For instance, after distributing \((q+3)\) over \((5q^2-15q+9)\), you end up with a longer expression like: \(5q^3-15q^2+9q+15q^2-45q+27\).

The goal now is to simplify, which involves collecting similar terms to reveal a cleaner form.

A simplified expression is more conducive to further mathematical operations or evaluations, helping clarify both the expression's structure and its significance. An easy step-by-step approach ensures you don't lose sight of the original problem's intention.

Combining Like Terms

Combining like terms is the final step in organizing an algebraic expression. This involves identifying terms in the expression that share the same powers of variables and then summing their coefficients.

• For example, \(5q^3-15q^2+9q+15q^2-45q+27\) contains terms that can be combined:
• \(-15q^2\) and \(15q^2\), which results in \(0q^2\), and can be dropped,
• \(9q\) and \(-45q\), which results in \(-36q\).

Combining like terms simplifies the polynomial effectively into fewer terms and a clearer expression like \(5q^3-36q+27\).

This step makes the expression far easier to interpret or solve in subsequent steps of an algebra problem, maintaining balance and reducing unnecessary work later on.