Problem 64 - Elementary Algebra

Question

Factor completely. Identify any prime polynomials. $$ 5 k y^{2}-10 y^{2} $$

Step-by-Step Solution

Answer

The factored form is \( 5y^2(k - 2) \). \( k - 2 \) is a prime polynomial.

Step 1: Identify the GCD

Determine the greatest common divisor (GCD) of the coefficients and the variables in the terms. In this case, both terms have a common factor of 5 and a common variable factor of \( y^2 \).

Step 2: Factor out the GCD

Factor out the GCD from the polynomial. The GCD is \( 5y^2 \). When you factor it out, the expression becomes: \[ 5y^2(k - 2) \]

Step 3: Check for prime polynomial

Check if the remaining factor \( k - 2 \) can be factored further. Since \( k - 2 \) is a linear polynomial without further factorization, it is a prime polynomial.

Key Concepts

Greatest Common Divisor (GCD)

When factoring polynomials, finding the Greatest Common Divisor (GCD) is a crucial first step. The GCD is the highest number that divides all terms of a polynomial exactly. In the polynomial given, \[ 5ky^2 - 10y^2 \], we identify the GCD by examining both the coefficients and the variable parts. For the coefficients, the numbers 5 and 10 share a GCD of 5. For the variables, the common factor is \( y^2 \). Hence, the GCD of the entire polynomial is \( 5y^2 \). This means that both terms can be divided by \( 5y^2 \).

Prime Polynomials

Once you factor out the GCD, you need to check if the resulting polynomial can be factored further. Let's look at the expression after factoring out the GCD:\[ 5y^2(k - 2) \]. Here, the factor inside the parentheses is \( k - 2 \). A polynomial is considered 'prime' when it cannot be factored further over the set of integers. In our case, \( k - 2 \) is a linear polynomial and cannot be factored any further. Thus, \( k - 2 \) is a prime polynomial. Identifying prime polynomials is important because it tells us that we have factored the polynomial completely.

Factoring Techniques

Factoring is a technique used to simplify polynomials by breaking them down into simpler 'factors' that, when multiplied together, give the original polynomial. Here are the steps to follow when factoring a polynomial, illustrated with our example:

• Identify the GCD: As previously discussed, find the greatest common factor of all terms. In our case, it was \( 5y^2 \).
• Factor out the GCD: Divide each term of the polynomial by the GCD and extract it. The polynomial becomes \( 5y^2(k - 2) \).
• Check for further factorization: Examine each factor to see if they can be factored more. For our example, \( k - 2 \) is a linear factor and prime.

These steps simplify the polynomial and make it easier to work with. Consistent practice with these techniques helps in identifying factorable and prime polynomials efficiently.