Problem 60 - A Graphical Approach to College Algebra

Question

Perform the indicated operations. $$2\left(3 r^{2}+4 r+2\right)-3\left(-r^{2}+4 r-5\right)$$

Step-by-Step Solution

Answer

The result is \(9r^2 - 4r + 19\).

Step 1: Distribute the first term

Begin by distributing the multiplication inside the first parentheses. You have the expression \(2(3r^2 + 4r + 2)\). Multiply 2 with each term inside the parentheses: \(2 \times 3r^2 = 6r^2\), \(2 \times 4r = 8r\), and \(2 \times 2 = 4\). So, the expression becomes \(6r^2 + 8r + 4\).

Step 2: Distribute the second term

Now, distribute the multiplication inside the second parentheses. You have the expression \(-3(-r^2 + 4r - 5)\). Multiply \(-3\) with each term inside the parentheses: \(-3 \times (-r^2) = 3r^2\), \(-3 \times 4r = -12r\), and \(-3 \times (-5) = 15\). So, the expression becomes \(3r^2 - 12r + 15\).

Step 3: Combine like terms

Add the resulting expressions from Steps 1 and 2 together: \((6r^2 + 8r + 4) + (3r^2 - 12r + 15)\). First, combine the \(r^2\) terms: \(6r^2 + 3r^2 = 9r^2\). Next, combine the \(r\) terms: \(8r - 12r = -4r\). Finally, combine the constant terms: \(4 + 15 = 19\). Thus, the expression simplifies to \(9r^2 - 4r + 19\).

Key Concepts

Distributive Property

The Distributive Property is a foundational concept in algebra that allows you to simplify expressions by distributing the multiplication across terms within parentheses. In our exercise,you start with the expression \(2(3r^2 + 4r + 2)\).
To apply the distributive property, multiply the 2 by each term inside the parentheses. This gives:

• \(2 \times 3r^2 = 6r^2\)
• \(2 \times 4r = 8r\)
• \(2 \times 2 = 4\)

Writing this out, the expression simplifies to\(6r^2 + 8r + 4\).
The same process is applied to the second part of the original problem, \(-3(-r^2 + 4r - 5)\). Distribute \(-3\) as follows:

• \(-3 \times (-r^2) = 3r^2\)
• \(-3 \times 4r = -12r\)
• \(-3 \times (-5) = 15\)

This results in the expression \(3r^2 - 12r + 15\). By applying the Distributive Property, you can manage complex polynomial expressions with ease.

Combining Like Terms

Once you have distributed the terms, the next step in polynomial simplification is to combine like terms. Like terms are terms that contain the same variables raised to the same power.
Looking at the two resulting expressions: \(6r^2 + 8r + 4\) and \(3r^2 - 12r + 15\), you can see several terms that are similar.
\(r^2\) terms are combined as follows:

• \(6r^2 + 3r^2 = 9r^2\)

\(r\) terms are combined by:

• \(8r - 12r = -4r\)

And the constants by:

• \(4 + 15 = 19\)

After combining these terms, the polynomial is simplified to \(9r^2 - 4r + 19\).
Combining like terms reduces the complexity of expressions, making them easier to handle and understand.

Polynomial Simplification

Polynomial Simplification involves using the distributive property and combining like terms to make a polynomial easier to work with or interpret.
Starting from the expression \(2(3r^2 + 4r + 2) - 3(-r^2 + 4r - 5)\), the process of simplification turns a potentially daunting expression into a streamlined one.
The steps are:

• Distribute any factors across all terms within parentheses.
• Combine any like terms to reduce the polynomial's complexity.

After applying these methods, you end up with \(9r^2 - 4r + 19\), which is much easier to interpret and use moving forward.
Understanding polynomial simplification not only helps solve problems quickly but also builds a deeper comprehension of algebraic structures and their behavior.