Problem 73 - Beginning and Intermediate Algebra

Question

Perform the indicated operations. $$\begin{array}{l} \left(2 t^{3}-8 t^{2}+t+10\right) \\ -\left[\left(5 t^{3}+3 t^{2}-t+8\right)+\left(-6 t^{3}-4 t^{2}+3 t+5\right)\right] \end{array}$$

Step-by-Step Solution

Answer

The short answer is the final expression after performing the indicated operations: \[3t^3 - 7t^2 - t - 3\]

Step 1: First, let's simplify the expression inside the brackets, which consists of two expressions separated by a plus sign. In order to simplify, use the associative property of addition and rearrange the terms. \[\left(5 t^{3}+3 t^{2}-t+8\right)+\left(-6 t^{3}-4 t^{2}+3 t+5\right)\] Adding the terms together, we have: \[5t^3 + 3t^2 - t + 8 - 6t^3 - 4t^2 + 3t + 5\] #Step 2: Combine like terms of the simplified expression inside brackets#

Now, combine the like terms in the simplified expression: \[(5t^3 - 6t^3) + (3t^2 - 4t^2) + (-t + 3t) + (8+5)\] This simplifies to: \[-t^3 - t^2+ 2t + 13\] Now, we have our simplified expression inside the brackets: \(-\left(-t^3 - t^2 + 2t + 13\right)\) #Step 3: Perform subtraction between two expressions#

Step 2: Now, perform the subtraction between the two expressions, \(2t^3 - 8t^2 + t + 10\) and -\(\left(-t^3 - t^2 + 2t + 13\right)\). \[\begin{array}{l} (2t^3 - 8t^2+ t+ 10) - \left(-t^3 - t^2 + 2t + 13\right) \end{array}\] To perform subtraction, distribute the negative sign to each term in the second expression: \[2t^3 - 8t^2+ t+ 10 + t^3 + t^2 - 2t - 13\] #Step 4: Combine like terms to get the final expression#

Combine the like terms in the expression: \[(2t^3 + t^3) + (-8t^2 + t^2) + (t - 2t) + (10 - 13)\] This simplifies to: \[3t^3 - 7t^2 -t - 3\] So, after performing the indicated operations, the final expression is: \(3t^3 - 7t^2 - t - 3\).

Key Concepts

Combining Like Terms

In algebra, we often work with polynomials that contain multiple terms. A fundamental concept here is combining like terms. To do this, first, identify the terms with the same variable and exponent. For example, in the expression \(-t^3 - t^2 + 2t + 13\), terms like \(5t^3\) and \(-6t^3\) are considered like terms because they both involve \(t^3\).

• To combine these like terms, simply add or subtract their coefficients.
• For instance, with \(5t^3 - 6t^3\), subtract \(6\) from \(5\) to get \(-t^3\).

By simplifying in this way, it makes it easier to manage and compute expressions. This is because you're reducing the number of terms you need to deal with overall. Remember, only those parts with the exact same variables and exponents can be combined together.

Associative Property

The associative property is one of the bedrock principles of algebra and arithmetic. When dealing with addition or multiplication, this property allows us to rearrange terms or numbers without changing the result.
Take, for instance, the expression: \((5t^3 + 3t^2) + (-6t^3 - 4t^2)\). The associative property tells us it's completely fine to regroup them as \( (5t^3 - 6t^3) + (3t^2 - 4t^2)\).

• This property is extremely useful as it gives flexibility in simplifying expressions, especially those with many terms.
• Whether you're adding numbers of single digits or working with complex polynomials, as long as the operation remains, the grouping does not affect the outcome.

Understanding and applying the associative property effectively can aid tremendously in solving algebraic problems with ease.

Distributive Property

The distributive property is crucial when dealing with expressions enclosed in parentheses, especially when a subtraction sign precedes them. This property states that a single term outside a set of parentheses can be multiplied (or distributed) across all terms within the parentheses.
An example from our exercise is the expression\(-(t^3 + t^2 - 2t - 13)\). Here, the negative sign functions like a coefficient of \(-1\).

• Distribute this \(-1\) across each term inside to get \(-t^3 - t^2 + 2t + 13\).
• This technique allows the simplification of complex expressions and makes subsequent operations manageable.

Employing the distributive property often reveals underlying relationships between terms, enabling combinations and simplifications that lead to more elegant solutions.