Problem 37 - Elementary Algebra

Question

For exercises 37-46, use a pattern to factor. Check. Identify any prime polynomials. $$ x^{3}+125 $$

Step-by-Step Solution

Answer

The factored form is \((x + 5)(x^2 - 5x + 25)\) with \(x^2 - 5x + 25\) being prime.

Step 1: Identify the type of polynomial

Recognize that the polynomial is a sum of cubes. The given polynomial is of the form \(x^3 + 125\), which can be written as \(x^3 + 5^3\).

Step 2: Use the sum of cubes formula

The sum of cubes formula is \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\). Here, \(a = x\) and \(b = 5\).

Step 3: Apply the formula

Substitute \(a = x\) and \(b = 5\) into the sum of cubes formula: \(x^3 + 5^3 = (x + 5)(x^2 - 5x + 25)\).

Step 4: Verify the factors

Check that the expression \((x + 5)(x^2 - 5x + 25)\) correctly factors back to \(x^3 + 5^3\) confirming the factorization is correct.

Step 5: Determine if either factor is a prime polynomial

The factor \(x + 5\) is a binomial and cannot be factored further. The trinomial \(x^2 - 5x + 25\) does not factor further easily and is considered a prime polynomial.

Key Concepts

Sum of Cubes

When working with polynomial expressions, one useful pattern to recognize is that of the sum of cubes. This specific type of polynomial takes the form \(a^3 + b^3\). Identifying a sum of cubes allows us to apply a specific factorization formula. For example, in the polynomial \(x^3 + 125\), we can see that it fits the sum of cubes form with \(a = x\) and \(b = 5\), since 125 is equal to \(5^3\).
To factor a sum of cubes, we use the formula:
\(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\).
By identifying the correct values for \(a\) and \(b\), we can plug them into the formula to achieve the correct factorization.

Prime Polynomial

A prime polynomial is a polynomial that cannot be factored any further over the integers. After factoring the sum of cubes in the example \(x^3 + 125\), we get two factors: \(x + 5\) and \(x^2 - 5x + 25\).
The first factor \(x + 5\) is a simple binomial and cannot be factored further.
The second factor \(x^2 - 5x + 25\) is a trinomial. To determine if this trinomial can be factored, we would look for two numbers that multiply to give 25 and add to give -5. Since no such pair of numbers exists, \(x^2 - 5x + 25\) is considered a prime polynomial.

Factorization Formula

The factorization formula we use for the sum of cubes is a powerful tool in algebra. It allows us to break down complex expressions into simpler factors. The formula states:
\(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\).
This can be remembered as a two-part factored form with one binomial and one trinomial.
When applying this formula to our given polynomial \(x^3 + 125\), we set up:
\[a = x\]
\[b = 5\]
and substitute these values into the formula to get:
\(x^3 + 5^3 = (x + 5)(x^2 - 5x + 25)\).
This process simplifies what initially appears to be a complex polynomial into a product of simpler polynomials that can help solve or analyze the polynomial further.