Problem 22 - Elementary Algebra

Question

(a) factor out the greatest common factor. Identify any prime polynomials. (b) check. $$ 70 y^{2}-70 y $$

Step-by-Step Solution

Answer

The factorized form is 70y(y - 1) and the polynomial y - 1 is prime.

Step 1: Identify the Greatest Common Factor (GCF)

First, identify the greatest common factor (GCF) of the coefficients and the variable. In this case, the coefficients are 70 and 70, and the variable is common. The GCF of 70 and 70 is 70, and the variable common to both terms is y.

Step 2: Factor out the GCF

Factor the GCF, 70y, out of each term: \[70y^{2} - 70y = 70y(y - 1)\].

Step 3: Check for Prime Polynomials

Determine whether the polynomial inside the parenthesis is prime. The polynomial \(y - 1\) has no common factors other than 1 and itself, indicating it is prime.

Step 4: Verification

Distribute back the GCF to ensure that the factorization is correct: \[70y(y - 1) = 70y^{2} - 70y.\] The original polynomial is obtained, confirming the factorization is correct.

Key Concepts

Greatest Common Factor (GCF)

When factoring polynomials, finding the Greatest Common Factor (GCF) is an essential first step. The GCF is the largest factor that two or more terms share. In this exercise, we work with the expression \(70y^2 - 70y\).
First, let's examine the coefficients and variables:

• The coefficients are 70 and 70.
• The variable part is \(y\).

The GCF of the coefficients is 70 because it is the largest number that can divide both coefficients. The variable part that is common to both terms is \(y\).
Putting it together, the GCF of the entire expression is \(70y\). Next, to factor out the GCF, we divide each term in the polynomial by the GCF and then factor it out:
\( 70y^2 - 70y = 70y(y - 1) \).
This step simplifies the original polynomial while retaining all the necessary information.

Prime Polynomial

A prime polynomial is a polynomial that cannot be factored into polynomials of lower degrees with integer coefficients. After factoring out the GCF, the remaining polynomial inside the parenthesis is \(y - 1\).
To determine if it's a prime polynomial, we check for any common factors other than 1.

In this expression:

• The coefficient of \(y\) is 1.
• The constant term is -1.

Since the only factors of 1 are 1 and -1, and the polynomial does not have any further common factors, we confirm that \(y - 1\) is indeed a prime polynomial. This implies it cannot be simplified any further through factoring.
Understanding prime polynomials is crucial because it limits how much you can break down a polynomial and indicates that you've reached the simplest form.

Polynomial Verification

Verification is the final step to ensure the correctness of your factorization. To verify the factorization of the polynomial \(70y(y - 1)\) from the original expression \(70y^2 - 70y\), we perform the distributive property.
Let's distribute \(70y\) across \(y - 1\):\ \( 70y(y - 1) = 70y^2 - 70y \).

When we expand this expression, we arrive back at our original polynomial.
This step ensures that the factorization was done correctly. Verification helps to avoid errors and confirms that the work is accurate. Always make it a habit to verify your factorized expressions, as it increases the reliability of your mathematical solutions.