Problem 66 - Elementary Algebra

Question

(a) factor by grouping. Identify any prime polynomials. (b) check. $$ 5 x^{2}-5 x z-x z^{2}-z^{3} $$

Step-by-Step Solution

Answer

The factored form is (x - z)(5x - z). Both factors are prime.

Step 1 - Group Terms

Group the terms of the polynomial in pairs: 5x^2 - 5xz - xz^2 - z^3.

Step 2 - Factor Common Terms from Each Pair

Factor out the greatest common factor (GCF) from each pair: 5x(x - z) - z^2(x + z).

Step 3 - Factor the Whole Expression

Look for a common binomial factor in the pairs: 5x(x - z) - z(x - z). The common binomial factor is x - z, so we can write the expression as: (x - z)(5x - z).

Step 4 - Check for Prime Polynomials

Check if (x - z) and (5x - z) can be factored further. Since these are already linear polynomials and cannot be factored further, they are both prime.

Step 5 - Verification

Expand the factored form to verify it equals the original polynomial. (x - z)(5x - z) = 5x^2 - xz - 5xz + z^2 = 5x^2 - 5xz - xz^2 - z^3. Thus, the factored form is correct.

Key Concepts

Grouping in Algebra

Grouping in algebra is a technique used to factor polynomials, especially when dealing with four or more terms. We group terms to make factoring easier.
For example, in the polynomial \(5x^2 - 5xz - xz^2 - z^3\), we can group the terms in pairs: \(5x^2 - 5xz\) and \(-xz^2 - z^3\). By pairing like this, we prepare the polynomial for the next steps in factoring.
This method works well when terms in each group have common factors. The idea is to reveal common binomial factors by handling smaller chunks first.

Greatest Common Factor

The greatest common factor (GCF) is the largest factor that divides two or more terms. In algebra, finding the GCF is a crucial step in simplifying expressions.
For instance, within our grouped pairs \(5x^2 - 5xz\) and \(-xz^2 - z^3\), we identify the GCF for each pair. The GCF of \(5x^2 - 5xz\) is \5x\, giving us \(5x(x - z)\).
The GCF of \(-xz^2 - z^3\) is \(-z^2\), leading to \(-z^2(x - z)\).
This extraction of common factors simplifies groups, making the next steps in factoring more straightforward.

Prime Polynomials

Prime polynomials are polynomials that cannot be factored further. They are the simplest form of a polynomial.
In our solution \((x - z)(5x - z)\), we must check if \(x - z\) and \(5x - z\) can be factored further. Since both are linear with no further factors, they are considered prime polynomials.
Understanding prime polynomials ensures we don't miss simpler forms of factorization and confirms we take the correct factor end results.

Verification of Factoring

Verification of factoring involves expanding the factored form to ensure it matches the original polynomial. This step confirms the accuracy of our factorization process.
Starting with \( (x - z)(5x - z)\), we expand to check:

• \ 5x^2 - xz - 5xz + z^2 \
• Combine like terms: \ 5x^2 - 5xz - xz^2 - z^3 \

Since this matches the original polynomial, our factorization is verified as correct. Always include this step to ensure your factorization is accurate.