Problem 63 - Elementary Algebra

Question

Factor completely. Identify any prime polynomials. $$ 3 a x^{2}-6 x^{2} $$

Step-by-Step Solution

Answer

The expression factors to \(3x^2(a - 2)\) and \(a - 2\) is a prime polynomial.

Step 1: Identify Common Factors

Look for common factors in each term of the polynomial. For the given polynomial, both terms, \(3ax^2\) and \(-6x^2\), share a common factor of \(3x^2\).

Step 2: Factor Out the Common Factor

Factor out the common factor \(3x^2\) from each term of the polynomial: \[3ax^2 - 6x^2 = 3x^2(a - 2)\]

Step 3: Check for Prime Polynomials

Examine the resulting binomial \(a - 2\). This expression cannot be factored further and is therefore a prime polynomial.

Key Concepts

Common Factors

In mathematics, common factors are numbers or expressions that divide exactly into two or more terms. When factoring polynomials, identifying and factoring out the common factors simplifies the polynomial. For instance, in the polynomial \(3ax^2 - 6x^2\), identifying \(3x^2\) as a common factor allows for easier simplification.
To find common factors:

• Examine the coefficients and variables in each term.
• Look for the greatest common divisor (GCD) among coefficients.
• Identify any shared variables and their lowest powers.

In our case, both \(3ax^2\) and \( -6x^2\) share the common factor \(3x^2\). By factoring it out, we get \[3x^2(a - 2)\], which is much simpler.

Prime Polynomials

Prime polynomials are polynomials that cannot be factored further over the set of integers. Identifying prime polynomials is crucial because it indicates that no further simplification can occur. Once we factor out common factors in a polynomial, we must check if the remaining polynomial can be factored any more.
Consider our example: \[3ax^2 - 6x^2 = 3x^2(a - 2)\]The binomial \(a - 2\) cannot be factored any further using integers. Therefore, it is a prime polynomial. Understanding this concept can save time and effort in solving polynomial problems effectively.

Binomials

Binomials are algebraic expressions containing two terms. Factoring binomials follows specific rules and patterns. In the example \(3ax^2 - 6x^2\), after factoring out the common factor \(3x^2\), we get the binomial \(a - 2\).
When working with binomials:

• Check if each term is a perfect square or cube.
• Use factoring formulas like difference of squares: \(a^2 - b^2 = (a + b)(a - b)\).
• Simplify by identifying and factoring out common terms first.

In our scenario, \(a - 2\) is already in its simplest form and does not follow any special factoring patterns. Recognizing when a binomial is prime is essential to completing the factoring process.