Problem 64 - A Graphical Approach to College Algebra

Question

Perform the indicated operations. $$-z^{3}(9-z)+4 z(2+3 z)$$

Step-by-Step Solution

Answer

The simplified expression is \[z^4 - 9z^3 + 12z^2 + 8z\].

Step 1: Expand the Expression

To perform the indicated operations, we first need to expand the terms in the expression. Start with the first part, \[-z^{3}(9-z)\].Apply the distributive property to get:\[-z^3 \cdot 9 + z^3 \cdot z = -9z^3 + z^4\].Next, expand the second part, \[4z(2+3z)\].Apply the distributive property to get:\[4z \cdot 2 + 4z \cdot 3z = 8z + 12z^2\].

Step 2: Combine Like Terms

Now, let's combine all the expanded terms:\[z^4 - 9z^3 + 12z^2 + 8z\].There are no like terms that can be combined further, so this is our simplified expression.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps simplify expressions. It allows you to multiply a single term by each term within a set of brackets. This can be particularly useful when dealing with polynomial expressions, as it facilitates the expansion of these expressions. For example, in the given exercise, we start with \(-z^3(9-z)\).
Using the distributive property, we distribute \(-z^3\) across each term inside the brackets:

• First, multiply \(-z^3\) by 9 to get \(-9z^3\).
• Next, multiply \(-z^3\) by \(z\) to get \(z^4\).

This results in the expanded expression \(-9z^3 + z^4\).
Similarly, apply the distributive property to the second part \(4z(2+3z)\).
Multiply \(4z\) by each term within the brackets:

• \(4z imes 2 = 8z\)
• \(4z imes 3z = 12z^2\)

The expanded result is \(8z + 12z^2\).
Using the distributive property correctly is key to transforming a complex expression into something more manageable.

Combining Like Terms

Once an expression is expanded using the distributive property, it's time to simplify it by combining like terms. Like terms are those terms that have the exact same variable raised to the same power. For instance, in the polynomial expression from the exercise \(z^4 - 9z^3 + 12z^2 + 8z\), we identify terms based on their degree:

• The term \(z^4\) is unique, as no other terms have the same power of \(z\).
• The term \(-9z^3\) also stands alone for the same reason.
• The term \(12z^2\) has no like terms, so it remains unchanged.
• Lastly, \(8z\) shares no commonality with any other term.

Since there are no like terms to combine further, the expanded expression is already simplified. This simplification step helps give a clearer overview and makes handling the polynomial easier in future calculations.

Polynomial Expansion

Polynomial expansion is the process of transforming a product of expressions into a sum or difference of terms. We achieve this by applying the distributive property and then simplifying the expression by combining like terms when possible. In the context of the exercise,
we start with \(-z^3(9-z)+4z(2+3z)\).
By applying the distributive property, we expand it into \(-9z^3 + z^4\) and \(8z + 12z^2\).
From here, it’s important to list and order terms according to their degree of \(z\) while ensuring to combine any like terms.
For instance, after expansion, we arranged our terms as follows: \(z^4 - 9z^3 + 12z^2 + 8z\), following the descending order in terms of powers of \(z\).
This orderly representation of a polynomial makes it easier to read and gives insight into the function's behavior, especially when graphing or performing further algebraic manipulations. Mastery of polynomial expansion aids not only in simplifying expressions but also in understanding complex algebraic constructs.