Problem 60 - Elementary Algebra

Question

Factor completely. Identify any prime polynomials. $$ 3 w^{2}-2700 $$

Step-by-Step Solution

Answer

The completely factored form is \(3(w - 30)(w + 30)\).

Step 1: Identify the GCD

Identify the greatest common divisor (GCD) of the coefficients in the polynomial. Here, the coefficients are 3 and 2700. The GCD is 3.

Step 2: Factor out the GCD

Factor out the GCD from the polynomial: \[ 3w^{2} - 2700 = 3(w^{2} - 900) \]

Step 3: Recognize a difference of squares

Notice that the expression inside the parentheses \(w^2 - 900\) is a difference of squares. Recall that \(a^2 - b^2 = (a - b)(a + b)\). Here, \(w^2\) is \(a^2\) and 900 is \(b^2\), with \(900 = 30^2\).

Step 4: Apply the difference of squares formula

Rewrite the expression using the difference of squares formula: \[ w^{2} - 900 = (w - 30)(w + 30) \]

Step 5: Substitute back into the factored form

Substitute the factored form back into the original equation: \[ 3(w^{2} - 900) = 3(w - 30)(w + 30) \]

Step 6: Verify if the polynomial is prime

\(3(w - 30)(w + 30)\) is the completely factored form. Since it is factored into linear binomials, the polynomial is not prime.

Key Concepts

greatest common divisor

When factoring polynomials, the first step is often to identify the Greatest Common Divisor (GCD) of the coefficients. The GCD is the largest number that divides all of the coefficients without leaving a remainder. For example, in the polynomial \(3w^2 - 2700\), the coefficients are 3 and 2700. The GCD is 3 because it is the largest number that evenly divides both 3 and 2700.

difference of squares

A difference of squares is a special polynomial form where two squared terms are subtracted. It has the general formula \(a^2 - b^2 = (a - b)(a + b)\). In our example, after factoring out the GCD, we are left with \(w^2 - 900\). Recognizing that 900 is a perfect square, specifically \(30^2\), allows us to rewrite it as \(w^2 - 30^2\). Using the difference of squares formula, we can factor it as \( (w - 30)(w + 30) \).

prime polynomial

A prime polynomial is a polynomial that cannot be factored into smaller polynomials with integer coefficients. In our example, we factored \(3w^2 - 2700\) completely to get \ 3(w - 30)(w + 30) \. Since it can be broken down into linear binomials, it is not a prime polynomial. A polynomial is only prime if, after all attempts, you can't factor it any further using real numbers.