Problem 33 - Elementary and Intermediate Algebra

Question

Factor each polynomial by factoring out the opposite of the GCF. $$ -6 x^{2}-3 x y $$

Step-by-Step Solution

Answer

The polynomial factors to \(-3x(2x + y)\).

Step 1: Identify the Greatest Common Factor (GCF)

The given polynomial is \(-6x^2 - 3xy\). Analyze each term to find the common factors. Both terms share a factor of \( -3x \), which is the greatest common factor.

Step 2: Factor Out the GCF

Now that we identified the GCF as \(-3x\), we need to factor it out from the polynomial. Thus, we write the polynomial as a product of \(-3x\) and another polynomial: \(-3x(2x + y)\).

Step 3: Verify the Factorization

Distribute \(-3x\) back into \(2x + y\) to ensure the expression was correctly factored. \(-3x(2x) = -6x^2\) and \(-3x(y) = -3xy\). The original polynomial \(-6x^2 - 3xy\) is recovered, confirming the factorization is correct and complete.

Key Concepts

Greatest Common Factor (GCF)

When dealing with polynomials, finding the Greatest Common Factor (GCF) is often the first step in simplifying an expression. The GCF is the largest factor that divides each term of the polynomial. In the polynomial \[-6x^2 - 3xy\],for example, we identify each part:.-6x^2 contains factors of -1, 6, and x.-3xy contains factors of -1, 3, x, and y.
-3 is a common number factor for both terms, and x is shared as a variable factor, making -3x the GCF.
Finding the GCF involves careful observation of both numerical coefficients and variable parts. It sets the groundwork for further operations, such as factoring or division, allowing us to break down complex polynomials.

Polynomial Division

Polynomial division is like the familiar process of number division but applied to algebraic expressions. It's frequently used in the process of simplifying polynomials, such as when factoring out the Greatest Common Factor (GCF). Specifically, following the example of\[-6x^2 - 3xy\],once we know the GCF is -3x, we divide each term of the polynomial by this factor:

• Divide \[-6x^2\] by \[-3x\] to get 2x.
• Divide \[-3xy\] by \[-3x\] to get y.

This results in the simplified expression of \[(2x + y)\]. Polynomial division is essential for rewriting complex expressions more simply, especially in factored form, helping us evaluate or further manipulate expressions more effectively.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations that represent a particular mathematical idea. In our example with \[-6x^2 - 3xy\], the expression describes a relation involving terms with variables x and y. Handling algebraic expressions often involves steps like factoring, simplifying, and evaluating.
Understanding and manipulating these expressions are core skills in algebra. Expressions like \[-6x^2 - 3xy\] can be factored or expanded, setting the stage for solving equations, modeling real-world situations, or simplifying complex problems.
By representing unknown values and operations symbolically, algebraic expressions form the basis for tackling more advanced topics like functions and calculus, and help develop problem-solving skills in mathematics.