Problem 42 - Introductory Algebra for College Students

Question

Now let's move on to factorizations that may require two or more techniques. Factor completely, or state that the polynomial is prime. Check factorizations using multiplication or a graphing utility. $$3 r^{2}-27 r$$

Step-by-Step Solution

Answer

The completely factorized form of the expression \(3 r^{2}-27 r\) is \(3r(r - 9)\)

Step 1: Identify Common Factor

The first step in factorizing an expression is to look for any common factor in the given terms. Here, both terms in the expression \(3 r^{2}-27 r\) share a common factor of \(3r\).

Step 2: Factorize the Expresssion

Once the common factor is identified, we factor it out of the expression. This leaves us with \(3r * (r - 9)\).

Step 3: Check the Factored Expression

Finally, as a check or verification step, we can multiply the factored expression back to its original form. If done properly, \(3r * (r - 9)\) should multiply back to \(3 r^{2}-27 r\).

Key Concepts

Understanding the Common Factor

When dealing with polynomials, one of the first steps in factorization is identifying whether the terms in the expression share any common factor. A common factor is a number or variable that divides each term in the polynomial exactly, without leaving a remainder. This means each term can be expressed as a multiple of this common factor. In the polynomial \(3r^2 - 27r\), the terms \(3r^2\) and \(-27r\) both include "\(3r\)" as a factor.

• 3: This number can be divided evenly into both \(3\) and \(27\).
• r: Present in both terms, allowing it to be factored out.

Finding a common factor simplifies the polynomial, making the next steps in factorization more straightforward. Recognizing these factors not only speeds up the process but assists in verifying your solutions.

Exploring Factoring Techniques

Once you've identified the common factor, the next step is to factor it out of the polynomial. This involves dividing each term in the polynomial by the common factor and then reorganizing to obtain a simplified expression. In our case here:\[3r^2 - 27r = 3r(r - 9)\]Several techniques can be applied depending on the complexity of the polynomial, such as:

• Greatest Common Factor (GCF): As demonstrated, this is the simplest method for expressions sharing a common factor.
• Grouping: Used in larger polynomials where parts can be simplified separately before finding a common factor.
• Difference of Squares: Specific to expressions like \(a^2 - b^2\), which can be easily factored as \((a-b)(a+b)\).

No matter which technique you choose, understanding the underlying structure of polynomials plays a crucial role in effective factorization.

The Importance of Checking Factorization

After factorizing a polynomial, it's essential to verify your result to ensure that it is correct. This not only confirms the accuracy of your work but also helps solidify your understanding of the factoring process. To check the factorization:

• Reverse Multiplication: Multiply the factored terms back together and check if the result equals the original polynomial. For \(3r(r - 9)\), multiplying gives: \(3r^2 - 27r\).
• Graphing Utility: Tools like graphing calculators can visually verify equivalence by showing the same graph for both forms of the equation.

This step is a powerful practice, allowing you to catch errors and improve your skills in polynomial factorization. Making it a habit can ensure accuracy and confidence in mathematical problem-solving. Remember, practice makes perfect!