Problem 92 - Elementary Algebra

Question

Factor completely. Identify any prime polynomials. $$ p^{3}+27 z^{3} $$

Step-by-Step Solution

Answer

(p + 3z)(p^2 - 3pz + 9z^2) and both polynomials are prime.

Step 1: Identify the form

Recognize that the given polynomial is a sum of cubes: \[ p^3 + 27z^3 \]

Step 2: Apply the sum of cubes formula

The formula for factoring a sum of cubes is: \[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]Here, identify \( a \) and \( b \):\[ a = p \]\[ b = 3z \]

Step 3: Substitute in the formula

Substitute \( a \) and \( b \) into the sum of cubes formula:\[ p^3 + 27z^3 = (p + 3z)(p^2 - p(3z) + (3z)^2) \]

Step 4: Simplify the expression

Simplify the terms inside the second parenthesis:\[ p^3 + 27z^3 = (p + 3z)(p^2 - 3pz + 9z^2) \]

Step 5: Identify prime polynomials

Check if the factors \( p + 3z \) and \( p^2 - 3pz + 9z^2 \) can be factored further. They cannot be factored further, thus both are prime polynomials.

Key Concepts

Polynomial Factorization

Polynomial factorization involves breaking down a polynomial into a product of simpler polynomials. This process is useful for solving equations, finding roots, and simplifying expressions.

To factor a polynomial effectively, recognize common patterns or formulas, such as the difference of squares or the sum of cubes. Understanding these patterns makes it easier to identify and apply the appropriate factorization method.

Sum of Cubes

The sum of cubes formula is essential for factoring certain polynomials. This formula is expressed as:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]

When given a polynomial like \[ p^3 + 27z^3 \, \] the goal is to rewrite it in the form of \[ a^3 + b^3. \] Identify the terms involved: here \[ a = p \] and \[ b = 3z. \] Substitute these into the formula to get
\[ p^3 + 27z^3 = (p + 3z)(p^2 - p(3z) + (3z)^2). \]
Simplify the expression inside the parentheses to complete the factorization:
\[ p^3 + 27z^3 = (p + 3z)(p^2 - 3pz + 9z^2). \]

Prime Polynomials

Prime polynomials are polynomials that cannot be factored further over the set of integers. They are the polynomial equivalent of prime numbers in arithmetic. After using the sum of cubes formula on \[ p^3 + 27z^3, \] we get two factors: \[ p + 3z \] and \[ p^2 - 3pz + 9z^2. \]

Check whether each of these factors can be simplified further. In this case, neither \[ p + 3z \] nor \[ p^2 - 3pz + 9z^2 \] can be factored again using integer coefficients, making both of them prime polynomials. This means our factorization is complete.