Problem 17 - Elementary Algebra

Question

(a) factor out the greatest common factor. Identify any prime polynomials. (b) check. $$ 56 x-35 z $$

Step-by-Step Solution

Answer

The factored form is 7(8x - 5z). 8x - 5z is a prime polynomial.

Step 1: Identify the Greatest Common Factor (GCF)

First, determine the GCF of the coefficients 56 and 35. The greatest common factor of 56 and 35 is 7.

Step 2: Factor out the GCF

Factor out 7 from each term in the expression. This gives: displaymath = 7(8x - 5z).

Step 3: Identify Prime Polynomials

Check if the resulting polynomial inside the parentheses (8x - 5z) can be factored further. Since 8x - 5z cannot be factored any further, it is a prime polynomial.

Step 4: Check the Factorization

Expand the factored form to check if it matches the original expression. (7)(8x - 5z) = 56x - 35z. The factorization is correct.

Key Concepts

Greatest Common Factor

The **Greatest Common Factor (GCF)** is the largest number that can exactly divide each of the coefficients in a polynomial. In this exercise, we need to consider the coefficients 56 and 35. To find the GCF of these two numbers, you first list the factors of each:

• Factors of 56: 1, 2, 4, 7, 8, 14, 28, 56
• Factors of 35: 1, 5, 7, 35

The largest number that appears in both lists is 7, which is the GCF. **Why is the GCF important?** Finding the GCF helps to simplify polynomials by factoring out the highest common number. This makes the polynomial easier to work with and is often the first step in more complex algebraic operations.

Prime Polynomial

A **Prime Polynomial** is a polynomial that cannot be factored into smaller polynomials using integer coefficients. In this exercise, once we factored out the GCF from 56x - 35z, we obtained the polynomial inside the parentheses:
**8x - 5z** Now, we need to check if **8x - 5z** can be factored further. Since there are no common factors other than 1 between the two terms and no way to factor it into smaller integers, 8x - 5z is a prime polynomial. **Identifying prime polynomials** can help you recognize when to stop factoring. If the polynomial cannot be factored any further, it simplifies the problem and the polynomial is considered fully factored.

Factoring Technique

The **Factoring Technique** used in this exercise is based on two fundamental steps:
First, you identify the greatest common factor (GCF) of the coefficients. For 56x - 35z, the GCF of 56 and 35 is 7. Then, you factor out the GCF from each term of the polynomial. *Step-by-Step:*

• Write the original polynomial: 56x - 35z
• Determine the GCF: 7
• Divide each term by the GCF: (56x ÷ 7) - (35z ÷ 7)
• Combine the results: 7(8x - 5z)

Once the GCF has been factored out, always check if the remaining polynomial can be factored further. This technique simplifies the polynomial and can help in solving equations.

Polynomial Factorization

**Polynomial Factorization** is the process of breaking down a polynomial into simpler components called factors. These factors, when multiplied together, give you the original polynomial. **Steps to Factorize a Polynomial:** 1. **Identify the GCF**: Find the greatest common factor of the coefficients. 2. **Factor out the GCF**: Divide each term by the GCF and write the polynomial in factored form. 3. **Check for further factorization**: Ensure the polynomial inside the parentheses cannot be factored further. 4. **Verify**: Expand the factored form to check if it matches the original polynomial. For 56x - 35z:

• Factor out the GCF: 56x - 35z becomes 7(8x - 5z)
• Check if 8x - 5z can be factored further - it cannot, so it is a prime polynomial.

Polynomial factorization simplifies complex polynomials and is essential in simplifying expressions for solving algebraic equations.