Problem 67 - Elementary Algebra

Question

Factor completely. Identify any prime polynomials. $$ 3 p-3 z^{2} $$

Step-by-Step Solution

Answer

3(p - z^2)

Step 1: Identify the Greatest Common Factor (GCF)

First, look for the greatest common factor (GCF) of the terms. In this expression, the terms are 3p and -3z^2. The common factor is 3.

Step 2: Factor Out the GCF

Next, factor out the GCF from both terms. The expression becomes: 3(p - z^2).

Step 3: Check for Further Factoring

Now, check if the polynomial inside the parentheses, p - z^2, can be factored further. In this case, p - z^2 cannot be factored further because it is already in its simplest form.

Step 4: Conclusion

Since the polynomial cannot be factored further, the expression 3(p - z^2) is fully factored. Thus, the final factored form of the given polynomial is 3(p - z^2).

Key Concepts

greatest common factor

The greatest common factor (GCF) is the largest number that can exactly divide each of the terms in a polynomial. Identifying the GCF is often the first step in factoring polynomials since it simplifies the expression. In our given problem, the terms are \(3p\) and \(-3z^2\). Both terms share a common factor of 3. By factoring out the GCF, the expression becomes simpler and easier to work with: \(3(p - z^2)\). Recognizing and factoring out the GCF is crucial as it lays the foundation for further simplification.

prime polynomial

A prime polynomial is a polynomial that cannot be factored further over the set of integers. It's similar to prime numbers in that it has no divisors other than 1 and itself. In our exercise, after factoring out the GCF, we are left with \(p - z^2\) inside the parentheses. To check if this polynomial is prime, we need to see if it can be factored further. Since \(p - z^2\) does not fit any standard factoring forms and cannot be decomposed into simpler polynomials, it is considered prime. Identifying prime polynomials is essential for verifying that an expression is fully factored.

factoring techniques

Factoring techniques are methods used to break down a polynomial into simpler components or factors. Some common techniques include:

• Finding the Greatest Common Factor (GCF)
• Factoring by grouping
• Factoring trinomials
• Difference of squares
• Sum/difference of cubes

In our example, the technique used is factoring out the GCF. We identified the GCF (which is 3) and pulled it out of the polynomial \(3p - 3z^2\). This left us with a simpler expression inside the parentheses: \(p - z^2\). Understanding different factoring techniques and when to use them is critical in simplifying complex polynomials and making algebraic expressions more manageable.