Problem 58 - Elementary Algebra

Question

(a) factor by grouping. Identify any prime polynomials. (b) check. $$ 2 f^{3}-f^{2} g+2 f-g $$

Step-by-Step Solution

Answer

The factorization is \((f^2 + 1)(2f - g)\). The polynomial \(f^2 + 1\) is prime.

Step 1: Group the terms

Group the polynomial into two pairs: \( (2f^3 - f^2g) + (2f - g) \)

Step 2: Factor out the common factors in each group

Factor out the greatest common factor (GCF) from each group: \( f^2(2f - g) + 1(2f - g) \)

Step 3: Factor by grouping

Notice that \(2f - g\) is a common factor: \( (f^2 + 1)(2f - g) \)

Step 4: Identify any prime polynomials

\(f^2 + 1\) is a prime polynomial as it cannot be factored further with real numbers.

Step 5: Check the factorization

Expand \((f^2 + 1)(2f - g)\) to verify it equals the original polynomial: \( (f^2 + 1)(2f - g) = f^2(2f - g) + 1(2f - g) = 2f^3 - f^2g + 2f - g \)

Key Concepts

Factoring by grouping

Factoring by grouping is a handy technique used for polynomials, especially when they have four or more terms. It's like a puzzle where you group terms to make factoring simpler. In our exercise, we start by grouping the polynomial \(2f^3 - f^2g + 2f - g\) into two pairs: \((2f^3 - f^2g) + (2f - g)\).

This step helps to break down the problem. Next, we factor out the greatest common factor (GCF) from each group, leading to: \(f^2(2f - g) + 1(2f - g)\).

Notice both groups now contain the common factor \((2f - g)\). By factoring this common factor out, we further simplify the polynomial to \((f^2 + 1)(2f - g)\).

Factoring by grouping can make seemingly complex polynomials much more manageable. Keep practicing grouping terms effectively to master this useful technique.

Prime polynomials

Prime polynomials are similar to prime numbers. They cannot be factored further using real numbers, and they don’t have any factors other than 1 and themselves. In our example, after factoring by grouping, we got \((f^2 + 1)(2f - g)\).

Here, \(f^2 + 1\) is considered a prime polynomial because it cannot be factored any further. To check if a polynomial is prime, try factoring it using common methods like looking for a GCF, using the quadratic formula, or applying special factorization formulas.

If none of these methods work, then the polynomial is prime. Recognizing prime polynomials is a key skill because it tells you when you can stop factoring.

Greatest common factor

Finding the greatest common factor (GCF) is crucial in polynomial factorization. The GCF is the highest number or algebraic term that divides each term in a polynomial without leaving a remainder.

In our problem, we identified \((2f^3 - f^2g)\) and \((2f - g)\) as groups to factor. For the first group \(2f^3 - f^2g\), the GCF is \(f^2\), leading to \(f^2(2f - g)\). The second group \(2f - g\) doesn't have a common factor other than 1 so it remains \(1(2f - g)\).

By factoring the GCF out of each group, we simplify the polynomial and can then re-factor the entire expression effectively. Getting good at finding the GCF can make complex problems much more manageable and pave the way to easily factor polynomials by grouping and other methods.