Problem 22 - Algebra 2

Question

Factor completely. If the polynomial is not factorable, write prime. $$ z^{3}+125 $$

Step-by-Step Solution

Answer

The factored form is \((z + 5)(z^2 - 5z + 25)\).

Step 1: Identify as a sum of cubes

The given expression is \( z^3 + 125 \). Recognize that both terms are cubes: \( z^3 \) is the cube of \( z \), and \( 125 \) is the cube of \( 5 \). Hence, the polynomial is a sum of cubes.

Step 2: Recall the sum of cubes formula

The formula for factoring the sum of cubes \( a^3 + b^3 \) is: \((a + b)(a^2 - ab + b^2)\). For our expression, \( a = z \) and \( b = 5 \).

Step 3: Apply the sum of cubes formula

Substitute \( a = z \) and \( b = 5 \) into the formula: \( (z + 5)(z^2 - 5z + 25) \). This gives the factored form of the polynomial.

Step 4: Confirm factorization

Expand the factored form \( (z + 5)(z^2 - 5z + 25) \) to verify that it equals the original polynomial \( z^3 + 125 \). If it does, the factorization is correct.

Key Concepts

Sum of Cubes

In algebra, understanding the sum of cubes is crucial for factoring certain types of polynomials. A sum of cubes describes any expression in the form \(a^3 + b^3\). This is because each term in the expression is a cube of some base number. For instance, in the expression \(z^3 + 125\), recognize that \(z^3\) is the cube of \(z\), and \(125\) is the cube of \(5\), since \(5^3 = 125\).
Identifying the structure of sum of cubes helps in easily applying the correct factoring formula. Remember, the general form of factoring a sum of cubes \(a^3 + b^3\) is utilizing the formula:
\[(a + b)(a^2 - ab + b^2)\]
Using this formula simplifies the process of breaking down cubic expressions into their factors.

Factoring Techniques

Factoring is a technique used to write a polynomial as a product of its simpler polynomials. When dealing with sums and differences of cubes, specialized formulas play a key role. These formulas are designed to simplify expressions into more manageable parts.

• For the sum of cubes \(a^3 + b^3\), the formula is \((a + b)(a^2 - ab + b^2)\).
• Each factor includes a binomial \((a + b)\) and a trinomial \((a^2 - ab + b^2)\).

To apply these techniques successfully, identify the cube root of each term. In our case, with \(z^3 + 125\), we determine \(a = z\) and \(b = 5\). Substituting these values into the formula results in \((z + 5)(z^2 - 5z + 25)\).
This technique is efficient and ensures the derived factors will multiply back to the original polynomial.

Algebraic Expressions

An algebraic expression is a combination of numbers, letters (representing variables), and operations (add, subtract, multiply, divide). These are foundational in algebra and allow for expressions to be manipulated in various ways, such as by factoring.
When factorizing polynomials like \(z^3 + 125\), recognizing the type of expression—whether it's a sum of cubes, a difference of squares, or another form—guides the choice of factorization strategy. In this context:

• The polynomial \(z^3 + 125\) is summed from individual terms which can be decomposed using algebraic rules.
• Recognizing these patterns simplifies resolving complex expressions into their simplest forms, like \((z + 5)(z^2 - 5z + 25)\),

which further aids in solving or simplifying broader algebraic problems.