Problem 37 - Elementary Algebra

Question

(a) factor out the greatest common factor. Identify any prime polynomials. (b) check. $$ 10 w^{3}-32 w^{2}+8 w $$

Step-by-Step Solution

Answer

The factored form is \(2w(5w^2 - 16w + 4)\), with \(5w^2 - 16w + 4\) being prime.

Step 1: Identify Greatest Common Factor (GCF)

Look at the coefficients (10, -32, 8) and find the greatest common factor. The GCF of these coefficients is 2. Additionally, each term has at least one factor of 'w'. Thus, the overall GCF is 2w.

Step 2: Factor out the GCF

Factor 2w out of each term: \[ 2w (5w^2 - 16w + 4) \]

Step 3: Check for Further Factorization

Examine the remaining quadratic polynomial \(5w^2 - 16w + 4\) to see if it can be factored further. Check if it can be factored into a product of two binomials. For this polynomial, it cannot be factored further with integer coefficients, so it is a prime polynomial.

Step 4: Confirm the Factored Form

Multiply the factors back together to ensure we get the original polynomial: \[ 2w (5w^2 - 16w + 4) = 10w^3 - 32w^2 + 8w \]. The original polynomial is correctly factored.

Key Concepts

Greatest Common Factor

Before factoring polynomials, the first step is to identify the Greatest Common Factor (GCF). Identifying the GCF simplifies the polynomial and makes further factorization easier. The GCF is the largest factor that is common to each term in the polynomial.

Consider the polynomial from the example: 10w^3 - 32w^2 + 8w. First, look at the numerical coefficients: 10, -32, and 8. The GCF of these numbers is 2 because it is the largest number that divides evenly into 10, -32, and 8.

Next, examine the variable part. Each term includes the variable 'w'. To find the GCF for this part, take the lowest power of the variable present in each term, which is w^1. Therefore, the GCF of the entire polynomial is 2w.

To factor out the GCF, divide each term by 2w: 10w^3 ÷ 2w = 5w^2
-32w^2 ÷ 2w = -16w
8w ÷ 2w = 4
Hence, factoring out the GCF from the original polynomial gives: 2w(5w^2 - 16w + 4).

Prime Polynomials

A prime polynomial is a polynomial that cannot be factored into the product of two lower-degree polynomials with integer coefficients. Identifying prime polynomials is crucial because this tells us when we can stop factoring.

For example, after factoring out the GCF in our polynomial, we get the quadratic polynomial: 5w^2 - 16w + 4. To determine if this is a prime polynomial, we check if it can be written as the product of two binomials.

Typically, we can factorize a quadratic polynomial of the form ax^2 + bx + c using methods like factoring by grouping, applying the quadratic formula, or trial and error with possible roots. However, if these methods show that no pair of factors can produce the middle term when multiplied, then the polynomial is considered prime.

In this case, after testing possible factors, we see that 5w^2 - 16w + 4 cannot be factored further over the integers. Therefore, it is a prime polynomial.

Polynomial Factorization

Polynomial factorization is a process that breaks down a polynomial into a product of simpler polynomials. It's a key technique in algebra for simplifying polynomials and solving polynomial equations.

Let’s revisit the original polynomial 10w^3 - 32w^2 + 8w. After identifying and factoring out the GCF, we get: 2w(5w^2 - 16w + 4)
Now, we need to check if the quadratic polynomial 5w^2 - 16w + 4 can be factored further. As determined earlier, it is a prime polynomial, which means it cannot be factored using integer coefficients.

To confirm that our factorization is correct, we multiply the factors: 2w * (5w^2 - 16w + 4) = 10w^3 - 32w^2 + 8w. Through this step-by-step process, we ensure that the polynomial is correctly factored.

Understanding polynomial factorization helps us simplify polynomial expressions, solve polynomial equations, and better understand the structure of polynomial functions. The key steps involve:

• Identifying the Greatest Common Factor (GCF)
• Factoring out the GCF from the polynomial
• Checking if the remaining polynomial can be factored further
• Multiplying the factors back to verify the original polynomial