Problem 71 - Elementary Algebra

Question

Factor completely. Identify any prime polynomials. $$ h^{2}+100 k^{2} $$

Step-by-Step Solution

Answer

The polynomial \( h^2 + 100k^2 \) is a prime polynomial.

Step 1: Identify the form

Recognize that the given polynomial is in the form of a sum of squares: \( h^2 + 100k^2 \)

Step 2: Check for common factors

There are no common factors between \( h^2 \) and \( 100k^2 \).

Step 3: Determine if factorable

The sum of squares, \( h^2 + 100k^2 \), cannot be factored using real numbers. It is considered a prime polynomial.

Key Concepts

sum of squares

The term 'sum of squares' refers to a polynomial that consists of the sum of two squared terms. In our example, we have:

• First term: \( h^2 \)
• Second term: \( 100k^2 \)

Typically, a sum like this cannot be simplified or factored using real numbers. This is different from the 'difference of squares', which can be further factored into the product of two binomials. For the sum of squares, there is no similar factorization method that applies. So, in our exercise, \( h^2 + 100k^2 \) remains in its simplest form.

prime polynomial

When a polynomial cannot be factored further using real numbers, it is often referred to as a 'prime polynomial'. Think of this similar to a prime number in arithmetic, which cannot be divided by any number other than 1 and itself without leaving a remainder.In our exercise, we determined that \( h^2 + 100k^2 \) cannot be factored further. This is because there are no common factors and no factorization rule applying to the sum of squares using real numbers. Thus, we label \( h^2 + 100k^2 \) as a prime polynomial.

common factors

Before attempting to factor a polynomial, it is essential to check if there are any common factors between the terms. A common factor is a number or variable that divides each term of the polynomial evenly.For the polynomial \( h^2 + 100k^2 \), let's examine each term separately:

• \( h^2 \)
• \( 100k^2 \)

After checking both terms, we find that they share no common factors. Once it is confirmed that there are no common factors, we can proceed to other factorization methods. For this exercise, however, as \( h^2 + 100k^2 \) cannot be factored further, it remains labeled as a prime polynomial.