Problem 82 - Beginning and Intermediate Algebra

Question

Factor completely. $$z^{3}-1000$$

Step-by-Step Solution

Answer

The completely factored expression is \((z - 10)(z^2 + 10z + 100)\).

Step 1: Identify a and b

Here, we have a difference of cubes: \(z^3 - 1000 = z^3 - 10^3\). So we can identify \(a = z\) and \(b = 10\).

Step 2: Apply the difference of cubes formula

Now we can apply the formula \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\). Substitute \(a = z\) and \(b = 10\): \((z^3 - 10^3) = (z - 10)(z^2 + z(10) + 10^2)\).

Step 3: Simplify

Now let's simplify the expression: \((z^3 - 10^3) = (z - 10)(z^2 + 10z + 100)\). So the completely factored expression is \((z - 10)(z^2 + 10z + 100)\).

Key Concepts

Difference of Cubes

A difference of cubes is a specific type of expression that takes the form \(a^3 - b^3\). In simpler terms, it's when you're subtracting one cubic term from another. Recognizing this pattern is key to factoring such expressions. The benefit of identifying a difference of cubes is that we can use a specific formula to make factoring easier. This formula is:

• \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\)

To apply this, you first identify \(a\) and \(b\). For example, in the expression \(z^3 - 10^3\), we see that \(a = z\) and \(b = 10\). Once identified, simply plug \(a\) and \(b\) into the formula. This step transforms a more complicated polynomial into simpler binomial and trinomial expressions.

Algebraic Expressions

An algebraic expression is a mathematical phrase that contains numbers, variables (like \(z\)), and arithmetic operations (like addition or subtraction). They can be quite simple or very complex. Understanding how to manipulate these expressions is crucial in math, especially when it comes to factoring.
In our exercise, \(z^3 - 1000\) is an algebraic expression. It's a polynomial, which specifically refers to expressions with multiple terms. When dealing with polynomials, especially when they have a recognizable pattern, we can apply special strategies like the difference of cubes formula.

• Pay attention to the powers and coefficients.
• Look out for patterns like squares or cubes.

Breaking algebraic expressions down into smaller parts can help you manage them more easily and solve them more effectively.

Polynomial Factoring Methods

Factoring polynomials is an essential skill in algebra. It refers to the process of breaking down a complex polynomial into simpler components or factors. These smaller pieces, when multiplied together, give back the original polynomial.
One efficient method for factoring is recognizing patterns, such as the difference of squares, perfect square trinomials, and difference of cubes. These patterns have formulas that make factoring faster and more straightforward.

• For \(z^3 - 1000\), we identified it as a difference of cubes and used the corresponding formula.
• Another method is to factor by grouping, but it didn't apply here as nicely as the difference of cubes formula did.

Recognizing these methods and knowing when to apply them will greatly enhance your math solving skills!