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Chapter 5: Problem 171

Factor each polynomial completely. $$z^{3}-125$$

Short Answer

Expert verified
\((z - 5)(z^2 + 5z + 25)\)

Step by step solution

01

Identify the type of polynomial

Recognize that the polynomial is a cubic expression in the form of a difference of cubes: \(z^3 - 125 = z^3 - 5^3\)
02

Apply the difference of cubes formula

Recall the difference of cubes formula: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\). Here, \(a = z\) and \(b = 5\).
03

Substitute into the formula

Replace \(a\) and \(b\) with \(z\) and \(5\) respectively: \((z - 5)(z^2 + 5z + 25)\)
04

Verify the factorization

Expand \((z - 5)(z^2 + 5z + 25)\) to ensure it matches the original expression: \((z - 5)(z^2 + 5z + 25) = z^3 + 5z^2 + 25z - 5z^2 - 25z - 125 = z^3 - 125\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Difference of Cubes
Recognizing the difference of cubes is crucial when factoring certain polynomials. The difference of cubes formula states that any expression of the form \(a^3 - b^3\) can be factored into \(a - b)(a^2 + ab + b^2)\). This is because when you multiply these two binomials, the terms involving \(a^2b\) and \(ab^2\) cancel out, leaving only the cubic terms and constants. For example, in the polynomial given \(z^3 - 125\), we can see it's written in the form \(a^3 - b^3\) with \(a = z\) and \(b = 5\). So, applying the formula here would give us \(z^3 - 5^3 = (z - 5)(z^2 + 5z + 25)\).
Cubic Expressions
Cubic expressions are polynomials where the highest degree term is a cube, such as \(z^3\). These can often be factored using specific formulas like the difference of cubes. To identify if a polynomial can be factored using the difference of cubes, check if it can be written as \(a^3 - b^3\). For example, a common cubic expression might be \(x^3 - 8\). Here, you can rewrite \(8\) as \(2^3\), yielding \(x^3 - 2^3\). Knowing these forms helps us in reducing complex polynomials into simpler products of binomials and trinomials.
Polynomial Factorization
Polynomial factorization is a method used to break down polynomials into products of simpler polynomials. By factoring polynomials, you simplify the expressions, making them easier to work with, especially for solving equations. For the polynomial \(z^3 - 125\), we recognize it as a special case of a difference of cubes. So, we use the formula \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\) to factor it into \( (z - 5)(z^2 + 5z + 25) \). Verification is a critical final step. You can confirm this factorization is correct if re-multiplying the factors returns you to the original polynomial: \( (z - 5)(z^2 + 5z + 25) = z^3 - 125 \).

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