Problem 31 - 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. $$6 x^{2}+8 x$$

Step-by-Step Solution

Answer

The completely factored form of the given polynomial \(6 x^{2} + 8 x\) is \(2x (3x + 4)\).

Step 1: Identify the common factor

In the given polynomial \(6 x^{2} + 8 x\), observe that both terms (6x² and 8x) have a common factor which is 2x.

Step 2: Factor out the common factor

Apply the greatest common factor factoring technique which requires to divide each term of the polynomial by the common factor identified in Step 1. Thus, the polynomial becomes \(2x (3x + 4)\)

Step 3: Check the factoring result

To make sure the factoring result is correct, distribute the factored expression to return to the original polynomial. If done correctly, distributing \(2x * (3x + 4)\) should yield the original polynomial \(6 x^{2} + 8 x\)

Key Concepts

Greatest Common Factor

When dealing with polynomials, the first step in factoring them efficiently is identifying the greatest common factor (GCF). The GCF is the largest factor that divides each term of the polynomial.

In the polynomial \(6x^2 + 8x\), both terms have coefficients and variables. By examining the coefficients, 6 and 8, their greatest common divisor is 2. Furthermore, since both terms contain the variable \(x\), we must also include the lowest power of \(x\) common in all terms. In this expression, \(x\) is present in each term, making \(x\) itself part of the GCF.

Thus, the GCF of the polynomial \(6x^2 + 8x\) is \(2x\). By identifying the GCF correctly, you can move on to simplifying the expression by factoring out this common factor.

Factoring Techniques

Once you've determined the GCF, you use factoring techniques to rewrite the polynomial. In our example, after finding the GCF as \(2x\), the next step involves dividing each term of the polynomial by \(2x\) and factoring it out.

This process is known as "factoring by grouping," where you take the common factor outside the parentheses. For the polynomial \(6x^2 + 8x\), dividing the first term \(6x^2\) by \(2x\) gives \(3x\), and dividing the second term \(8x\) by \(2x\) gives 4.

Hence, the polynomial expression \(6x^2 + 8x\) can be factored to \(2x(3x + 4)\). This expression inside the parentheses, \(3x + 4\), is the result of our factoring. Factoring makes complex equations simpler and provides insight into their structure.

Checking Factoring Results

After factoring, it's crucial to check your work. Ensuring the correctness of your factoring can be done by redistributing the terms. Distributing essentially reverses the factoring process to verify if you will get the original polynomial back.

To check \(2x(3x + 4)\), distribute \(2x\) to each term inside the parentheses:

• Multiply \(2x \times 3x\) to get \(6x^2\).
• Multiply \(2x \times 4\) to get \(8x\).

Combining these terms, you end up with \(6x^2 + 8x\), which matches the original polynomial. This confirms that the factoring was executed correctly.

Checking doesn't just ensure accuracy; it deepens your understanding of how expressions are composed and decomposed, enhancing your algebraic intuition.