Problem 39 - Elementary Algebra

Question

(a) factor out the greatest common factor. Identify any prime polynomials. (b) check. $$ p^{4}-p^{3}-p^{2}-p $$

Step-by-Step Solution

Answer

Factor: \[p(p^{3} - p^{2} - p - 1)\], prime: \[p^{3} - p^{2} - p - 1\]

Step 1: Identify the Greatest Common Factor (GCF)

Examine all the terms: \(p^{4}, -p^{3}, -p^{2}, -p\). The GCF of all these terms is \(p\).

Step 2: Factor Out the GCF

Factor out \(p\) from each term: \[p(p^{3} - p^{2} - p - 1) \]

Step 3: Identify Prime Polynomials

Check if \(p^{3} - p^{2} - p - 1\) can be factored further. It cannot be factored further as it does not have any factors without using complex numbers, hence it is a prime polynomial.

Step 4: Verify by Multiplication

Multiply back to ensure factoring is correct: \[p(p^{3} - p^{2} - p - 1) = p^{4} - p^{3} - p^{2} - p\]. The original polynomial is recovered, confirming that the factoring is correct.

Key Concepts

Greatest Common Factor

The Greatest Common Factor (GCF) is the highest factor that divides all terms in a polynomial without leaving a remainder. In this exercise, we need to identify the GCF for the polynomial \(p^4 - p^3 - p^2 - p\).

To find the GCF, examine each term in the polynomial:

• \(p^4\)
• \(-p^3\)
• \(-p^2\)
• \(-p\)

Each term includes at least one \(p\). Therefore, the GCF is \(p\). Once identified, we factor the GCF from each term, transforming the polynomial into:
\[ p(p^3 - p^2 - p - 1) \]
This extraction is the first and essential step in simplifying or factoring polynomials.

Prime Polynomials

Prime polynomials are polynomials that cannot be factored into polynomials of lower degrees with integer coefficients (or without using complex numbers). In this exercise, after factoring out the GCF from \(p^4 - p^3 - p^2 - p\), we are left with:

\[ p^3 - p^2 - p - 1 \]
To determine if this expression is a prime polynomial, we attempt to factor it further. However, in this case, \(p^3 - p^2 - p - 1\) cannot be factored further using simple integer coefficients. Since no simpler factoring is possible, \(p^3 - p^2 - p - 1\) is considered a prime polynomial. Recognizing prime polynomials is crucial as it helps identify when we've simplified an expression as much as possible.

Verification by Multiplication

After factoring a polynomial, it's important to verify the correctness of the process by multiplying the factors back together. This ensures the original polynomial is recovered, confirming that no errors were made.

For our example, we factored \(p^4 - p^3 - p^2 - p\) into:
\[ p(p^3 - p^2 - p - 1) \]
To verify, we multiply the factored terms:

\[ p(p^3 - p^2 - p - 1) \]
Expanding this, we get:

• \(p \times p^3 = p^4\)
• \(p \times -p^2 = -p^3\)
• \(p \times -p = -p^2\)
• \(p \times -1 = -p\)

Adding all these terms together gives us:
\[ p^4 - p^3 - p^2 - p \]
This matches the original polynomial, verifying that our factoring was correct. Always perform verification to ensure reliability in your results.