Problem 33 - Algebra and Trigonometry

Question

Find the sum, difference, or product. \(r\left(r^{2}-9\right)+3 r^{2}(2 r-1)\)

Step-by-Step Solution

Answer

The simplified form is \( 7r^3 - 3r^2 - 9r \).

Step 1: Expand the Products

The expression to solve is: \( r(r^2 - 9) + 3r^2(2r - 1) \). First, expand each product separately. For \( r(r^2 - 9) \):- Multiply \( r \) by \( r^2 \), giving \( r^3 \).- Multiply \( r \) by \(-9 \), yielding \(-9r \).Therefore, \( r(r^2 - 9) = r^3 - 9r \). For \( 3r^2(2r - 1) \):- Multiply \( 3r^2 \) by \( 2r \), giving \( 6r^3 \).- Multiply \( 3r^2 \) by \(-1 \), yielding \(-3r^2 \).Therefore, \( 3r^2(2r - 1) = 6r^3 - 3r^2 \).

Step 2: Combine Like Terms

After expanding, we have:\( r^3 - 9r \) and \( 6r^3 - 3r^2 \).Combine these terms to form a single polynomial:- Combine \( r^3 \) terms: \( r^3 + 6r^3 = 7r^3 \).- Combine \(-3r^2 \) term (since there's no other \( r^2 \) term): it remains \(-3r^2 \).- Combine \(-9r \) term (since there's no other \( r \) term): it remains \(-9r \).The combined expression is: \( 7r^3 - 3r^2 - 9r \).

Step 3: Write the Final Expression

Finally, ensure that the expression is simplified and written in descending powers of \( r \):\[ 7r^3 - 3r^2 - 9r \].

Key Concepts

Expanding Expressions

In algebra, expanding expressions is a crucial step in solving complex polynomial problems. When we talk about expansion, we're looking to remove parentheses by using the distributive property. This means that each term inside the parenthesis is multiplied by the term outside.
By expanding, we're spreading out the terms to make them more manageable for further operations like combining. Let’s consider the example:
\( r(r^2 - 9) + 3r^2(2r - 1) \). This is a classic case where expansion is necessary.

• For \( r(r^2 - 9) \), we multiply \( r \) by each term inside the parenthesis: \( r imes r^2 \) and \( r imes -9 \). The resulting expanded terms are \( r^3 \) and \(-9r \).
• For \( 3r^2(2r - 1) \), distribute \( 3r^2 \) over \( 2r \) and \(-1 \), resulting in \( 6r^3 \) and \(-3r^2 \).

This method makes it easier to handle each component separately and prepares us for the next steps of polynomial simplification.

Combining Like Terms

Once expressions are expanded, combining like terms is the strategy used to simplify them further. Like terms are those that have identical variables raised to the same power. By combining them, we essentially condense the expression.
In our example, after expansion, we have: \( r^3 - 9r + 6r^3 - 3r^2 \). The key is to group and add or subtract these like terms:

• Identify like terms: Here, \( r^3 \) and \( 6r^3 \) are like terms since they both have the variable \( r \) raised to the power of 3. Similarly, \( -3r^2 \) and \( -9r \) stand alone as separate like terms due to their unique powers.
• Combine the like terms: Add \( r^3 + 6r^3 \) to get \( 7r^3 \). The terms \( -9r \) and \( -3r^2 \) remain unchanged as they have no like pairs in this expression.

Ultimately, by combining these terms, we achieve a more simplified expression: \( 7r^3 - 3r^2 - 9r \). This reduces clutter and makes the polynomial easier to interpret.

Simplifying Polynomials

Simplifying polynomials involves reducing the expression to its simplest form without changing its value, by combining all possible like terms and arranging them in a standard format. We aim to make the polynomial expression as concise as possible.
The final step in our process is to ensure the polynomial is written in descending powers, which is the conventional format:

• First, order the terms by decreasing powers of \( r \). In our simplified expression \( 7r^3 - 3r^2 - 9r \), the terms are appropriately ordered from highest power to lowest: \( r^3 \), \( r^2 \), and \( r \).
• Check for completeness and consistency. Ensure that there are no remaining like terms that need combining and nothing more can be simplified.

This process not only organizes the polynomial in a clear format but verifies that it is as simplified as possible, making further operations straightforward and error-free.