Problem 23 - Algebra 2 and Trigonometry

Question

Perform the indicated operations and write the result in simplest form. \((z-2)^{3}\)

Step-by-Step Solution

Answer

The simplest form is \(z^3 - 6z^2 + 12z - 8\).

Step 1: Understand the Expression

We are given the expression \((z-2)^3\). This means we need to multiply \((z-2)\) by itself three times: \((z-2) \times (z-2) \times (z-2)\).

Step 2: Multiply Two Terms

First, we need to compute \((z-2) \times (z-2)\). Apply the distributive property (a.k.a. FOIL method for binomials): \[(z-2) \times (z-2) = z \times z - z \times 2 - 2 \times z + 2 \times 2\]Simplifying, we get:\[= z^2 - 2z - 2z + 4\]Combine like terms:\[= z^2 - 4z + 4\]

Step 3: Multiply Result with Third Term

Next, we multiply the result from step 2 with the third \((z-2)\):\[(z^2 - 4z + 4) \times (z-2)\]Apply the distributive property:\[z^2 \times (z-2) = z^3 - 2z^2\]\[-4z \times (z-2) = -4z^2 + 8z\]\[4 \times (z-2) = 4z - 8\]Combine all these results:\[z^3 - 2z^2 - 4z^2 + 8z + 4z - 8\]

Step 4: Combine Like Terms

Combine all like terms from the expanded expression:Combine the \(z^2\) terms:\(-2z^2 - 4z^2 = -6z^2\)Combine the \(z\) terms:\(8z + 4z = 12z\)The final expression:\[z^3 - 6z^2 + 12z - 8\]

Step 5: Write the Result in Simplest Form

Since the expression \(z^3 - 6z^2 + 12z - 8\) has no like terms to combine further, it is already in its simplest form.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in mathematics that allows you to multiply a single term by each term within a bracket.
This property is particularly useful when dealing with polynomial expressions. It helps simplify expressions and makes calculations manageable.
Let's imagine a scenario with \((a + b)(c + d)\), where you distribute one element from the first pair to each element in the second:

• First, multiply \(a\) by \(c\)
• Next, multiply \(a\) by \(d\)
• Then, multiply \(b\) by \(c\)
• Lastly, multiply \(b\) by \(d\)

With these steps, the expression becomes: \(ac + ad + bc + bd\).
This approach is not just limited to numbers, it's equally applicable to algebraic expressions.
By applying the distributive property, we can ensure that each part of the expression is accounted for, leading to an accurate simplification.

FOIL Method

For polynomial operations, especially when multiplying two binomials, the FOIL method is a specific application of the distributive property.
FOIL stands for First, Outside, Inside, and Last.
These terms correspond to how you multiply the components of two binomials:

• **First:** Multiply the first term in each binomial.
• **Outside:** Multiply the outer terms of the binomials.
• **Inside:** Multiply the inner terms.
• **Last:** Multiply the last term in each binomial.

For example, consider the expression \((x + 2)(x - 3)\):

• **First:** \(x \times x = x^2\)
• **Outside:** \(x \times (-3) = -3x\)
• **Inside:** \(2 \times x = 2x\)
• **Last:** \(2 \times (-3) = -6\)

Combine these results to get the expanded form: \(x^2 - 3x + 2x - 6\).
The FOIL method helps ensure no crucial component is missed, leading to the correct and complete expansion of the expression.

Simplifying Expressions

Simplifying expressions involves combining the results of polynomial operations into a more compact and manageable form.
After distributing terms or using the FOIL method, you're often left with a long expression.
By simplifying, you make the expression easier to work with by removing any unnecessary complexity.
To simplify an expression, focus on combining like terms.
For example, take the expanded result from multiplying two binomials: \(x^2 - 3x + 2x - 6\).
Here, \(-3x\) and \(+2x\) are like terms and can be combined to \(-x\), leading to a simplified expression of \(x^2 - x - 6\).
Simplifying is crucial because it gives you a clearer view of the expression's structure and makes it easier to proceed with further calculations if necessary.

Combining Like Terms

Combining like terms is a crucial aspect of simplifying polynomial expressions.
But what exactly are like terms?
These are terms that contain the same variables raised to the same power.
When you combine like terms, you simply add or subtract the numerical coefficients while maintaining the common variable part.
This step helps in refining the expression.
For instance, consider the expression: \(-2z^2 - 4z^2 + 8z + 4z\).
First, identify terms with the same variable and power, then combine them:

• \(-2z^2\) and \(-4z^2\) combine to form \(-6z^2\)
• \(8z\) and \(4z\) combine to form \(12z\)

After combining, the expression becomes \(-6z^2 + 12z\).
This process is vital as it reduces the length of polynomial expressions and simplifies computations.