Problem 70 - Beginning and Intermediate Algebra

Question

Perform the indicated operations. $$\begin{aligned} &\left(21 r^{3}-8 r^{2}+3 r+2\right)+\left(-4 r^{2}+5\right)\\\ &-\left(6 r^{3}-r^{2}-4 r\right) \end{aligned}$$

Step-by-Step Solution

Answer

The simplified result after performing the indicated operations is \(15r^3 - 11r^2 + 7r + 7\).

Step 1: Observe the given expression

Firstly, we need to notice that the first operation is between the first and second polynomials that are written inside parentheses. Let's write this operation as: $$\begin{aligned} (21 r^{3}-8 r^{2}+3 r+2)+(-4 r^{2}+5) \end{aligned}$$

Step 2: Remove parentheses and combine like terms

Since it is an addition operation between the two polynomials, we can remove the parentheses directly. After removing parentheses, combine the like terms in the expression. $$\begin{aligned} 21 r^{3} - 8r^{2} + 3r + 2 - 4r^{2} + 5 \end{aligned}$$

Step 3: Combine like terms for the first two polynomials

Now, merge the like terms to simplify the expression further. $$\begin{aligned} 21 r^{3} - 12r^{2} + 3r + 7 \end{aligned}$$

Step 4: Write the final expression to be simplified

Next, we have to subtract the third polynomial from the simplified result we got in step 3. $$\begin{aligned} (21 r^{3}-12r^{2}+3r+7)-(6r^{3}-r^{2}-4r) \end{aligned}$$

Step 5: Remove parentheses and combine like terms

Since it is a subtraction operation between the resulting expression from step 3 and the third polynomial, we have to change the signs of the terms in the third polynomial before removing parentheses. Then, combine the like terms. $$\begin{aligned} 21 r^{3} - 12r^{2} + 3r + 7 - 6r^{3} + r^{2} + 4r \end{aligned}$$

Step 6: Combine like terms for the final expression

Finally, merge the like terms in the expression to get the simplified result. $$\begin{aligned} 15 r^{3} - 11r^{2} + 7r + 7 \end{aligned}$$ The result of the given operations is \(15r^3 - 11r^2 + 7r + 7\).

Key Concepts

Combining Like Terms

When working with polynomials, combining like terms is an essential step to simplify expressions. Like terms are terms that have the same variable raised to the same power. Understanding how to group these terms helps in simplifying the polynomial efficiently.

To combine like terms:

• Identify terms that have the same variable and exponent. For instance, in the expression \(21r^3 - 8r^2 + 3r + 2 - 4r^2\), the terms \(-8r^2\) and \(-4r^2\) are like terms because they both involve \(r^2\).
• Add or subtract the coefficients of these like terms. In the example above, you combine \(-8r^2\) and \(-4r^2\) to get \(-12r^2\).

By methodically identifying and combining like terms, you transform a complex polynomial into a simpler expression that is easier to manage and understand.

Addition and Subtraction of Polynomials

Addition and subtraction are fundamental operations you perform on polynomials to simplify or find a solution. Both involve handling parentheses properly and ensuring correct operations on each term.

Here’s a step-by-step approach:

• **Addition:** Simply group like terms from both polynomials and add their coefficients. Consider the expression \((21r^{3} - 8r^{2} + 3r + 2) + (-4r^{2} + 5)\). You remove the parentheses and get \(21r^{3} - 8r^{2} + 3r + 2 - 4r^{2} + 5\). Then, combine the like terms, \(-8r^{2}\) and \(-4r^{2}\), to continue with the process.
• **Subtraction:** Be mindful that subtraction involves changing the sign of every term from the polynomial being subtracted. This step is crucial. For example, in \((21r^{3} - 12r^{2} + 3r + 7) - (6r^{3} - r^{2} - 4r)\), you change the signs to get \(21r^{3} - 12r^{2} + 3r + 7 - 6r^{3} + r^{2} + 4r\).

Understanding these operations makes working with polynomials more intuitive and decreases chances of mistakes during calculations.

Simplifying Expressions

Simplifying expressions involves rewriting them in their simplest form. This means you need to combine like terms, resolve any operations like addition or subtraction, and ensure coefficients are combined wherever possible.

To simplify an expression such as our original polynomial operations, follow these steps:

• **Remove Parentheses:** Start by removing any parentheses, keeping in mind appropriate sign changes if subtraction is involved.
• **Combine Like Terms:** Next, group and simplify terms that have the same variable raised to the same power. For example, \(21r^{3}\) and \(-6r^{3}\) simplify to \(15r^{3}\).
• **Final Expression:** Once all similar terms are combined, rewrite the polynomial in a neat and simplified form such as \(15r^{3} - 11r^{2} + 7r + 7\).

Simplifying expressions properly helps in better problem-solving and a clearer understanding of polynomial operations. It reduces the polynomial to its most manageable form, making further calculations or evaluations straightforward.