Problem 28 - Introductory Algebra for College Students

Question

Factor completely, or state that the polynomial is prime. $$5 x^{2}-45$$

Step-by-Step Solution

Answer

The factorized form of the polynomial \(5x^2-45\) is \(5(x - 3)(x + 3)\).

Step 1: Identify the Common Factor

Observe the terms of the polynomial to identify if they share a common factor. In this case, \(5x^2\) and \(45\) both can be divided evenly by 5.

Step 2: Factorize the Polynomial

Factor out the common factor, 5. The expression will become \(5(x^2 - 9)\).

Step 3: Identify the Difference of Squares

The expression in the parentheses, \(x^2 - 9\), is a difference of squares. It can be factored into \((x - 3)(x + 3)\). Here, \(x^2\) is the square of x and \(9\) is the square of 3.

Step 4: Apply the Difference of Squares

Apply the difference of squares formula. The expression \(x^2 - 9\) transforms into \((x - 3)(x + 3)\).

Step 5: Write the Complete Factorized Form

Combining the step 2 and 4 results, the complete factored form of the given polynomial is \(5(x - 3)(x + 3)\).

Key Concepts

Common Factor

When we look to factor a polynomial, one of the first steps is to look for a common factor. This is a number or algebraic expression that evenly divides all terms of the polynomial. In the given exercise, both terms in the polynomial, 5x^2 and 45, have a common factor of 5.

Finding the common factor is essential as it simplifies the polynomial, making the next steps of factoring easier. It's akin to reducing a fraction to its lowest terms before performing operations with it. To extract the common factor, we divide each term by 5, which is the first step toward making the problem more manageable.

Difference of Squares

The difference of squares is an important concept in algebra where a binomial takes the form of a^2 - b^2. This special form can be factored into the product of two conjugates: (a - b)(a + b).

In the exercise provided, after factoring out the common factor, we are left with x^2 - 9, which is indeed a difference of squares since x^2 is the square of x, and 9 is the square of 3. By recognizing this pattern, we can apply the corresponding factoring rule to simplify the expression further into (x - 3)(x + 3).

Polynomial Factorization

Factoring polynomials is the process of breaking down a polynomial expression into a product of simpler polynomials. The goal is to express the polynomial as a product of the least possible number of irreducible factors.

Polymerization builds complex structures from simpler units, whereas polynomial factorization is like taking apart a complex structure to examine the simpler building blocks from which it's made. In our exercise example, the process began by extracting the common factor and proceeded to utilize the difference of squares rule to break down the polynomial into the factors 5, (x - 3), and (x + 3).

Introductory Algebra

In introductory algebra, students begin to explore the language of mathematics through the manipulation of symbols and letters to represent numbers and relationships between them. Concepts such as finding common factors, factoring polynomials, and understanding differences of squares are fundamental.

Understanding these concepts is critical for solving equations, graphing functions, and analyzing mathematical relationships. The exercise we examined showcases how various introductory algebraic techniques come together to solve polynomial equations, strengthening the foundation upon which more complex algebraic concepts are built.