Problem 23 - Beginning and Intermediate Algebra

Question

Perform the indicated operations and simplify. $$-\left(10 r^{3}-14 r+27\right)+3\left(3 r^{3}-13 r^{2}-15 r+6\right)$$

Step-by-Step Solution

Answer

The simplified expression after performing the indicated operations is \(-r^{3} - 39 r^{2} - 31 r - 9\).

Step 1: Distribute the negative sign and the 3

In this step, we distribute the negative sign across the first bracket and the 3 across the second bracket. \(-\left(10 r^{3}-14 r+27\right) = -10 r^{3} +14 r -27\) \(3\left(3 r^{3}-13 r^{2}-15 r+6\right) = 9 r^{3} -39 r^{2} -45 r + 18\) Now our expression becomes: \[ -10 r^{3} +14 r -27 + 9 r^{3} -39 r^{2} -45 r + 18\]

Step 2: Combine like terms

In this step, we will add or subtract the like terms, which are the terms with the same variables raised to the same power: Combine the \(r^3\) terms: \(-10 r^{3} + 9 r^{3} = -1 r^{3}\) Combine the \(r^2\) terms: \(-39 r^{2}\) (there's only one \(r^2\) term, so we don't need to do any operation) Combine the r terms: \(14 r - 45 r = -31 r\) Combine the constant terms: \(-27 + 18 = -9\) Now the expression becomes: \[ -1 r^{3} - 39 r^{2} - 31 r - 9\]

Step 3: Write the simplified expression

Now that we have combined all the like terms, our simplified expression is: \(-r^{3} - 39 r^{2} - 31 r - 9\)

Key Concepts

Combining Like Terms

When simplifying polynomial expressions, one crucial step is combining like terms. Like terms are terms that have the same variables raised to the same power. This means we can group them together in calculations. For example, in the polynomial

• -10r^3,
• 9r^3,
• -14r,
• -45r,
• -27,
• 18.

We identify -10r^3 and 9r^3 as like terms and combine them to get -1r^3.

Similarly, for linear terms, -14r and -45r combine to become -31r after addition. Constant terms like -27 and 18 also combine simply by adding or subtracting.
Combining like terms helps reduce the expression to its simplest form, making it easier to work with or solve questions.

Distributive Property

The distributive property is a fundamental technique in algebra used to simplify expressions. This property allows one to multiply a single term by each term inside a parenthesis. Consider a simple expression like

• -\((10r^3-14r+27)\)

by applying the distributive property, each term inside the parentheses is multiplied by -1 (due to the negative sign outside), giving us -10r^3 + 14r - 27.

Similarly, we distribute 3 over

• 3\((3r^3-13r^2-15r+6)\),

multiplying 3 with each term inside, resulting in 9r^3 - 39r^2 - 45r + 18.
This method reforms the expression outside of the parentheses and is especially helpful when dealing with complex or layered expressions.

Simplifying Expressions

The final goal in most algebraic operations is to simplify expressions to their most compact form. This involves performing all possible operations and reducing the expression:

• Combine all like terms
• Distribute any multipliers across terms
• Resolve any constants

As seen in our exercise, the expression is initially given as -\((10r^3-14r+27) + 3(3r^3-13r^2-15r+6)\).

By using the distributive property, simplifying the new expression -10r^3 + 14r - 27 + 9r^3 - 39r^2 - 45r + 18,
with combined like terms, we arrive at -r^3 - 39r^2 - 31r - 9.
This is crucial as a simplified expression can be more easily evaluated, understood, or used in further calculations.