Problem 9 - Elementary Algebra

Question

Factor out the GCF. $$ 9 x 3 y 2-18 x 2 y 2+27 x y 2 $$

Step-by-Step Solution

Answer

The GCF is factored as \(9xy^2(x^2 - 2x + 3)\).

Step 1: Identify the common factors

Examine each term for common factors. The expression is \(9x^3y^2 - 18x^2y^2 + 27xy^2\). Identify common variables and coefficients. The coefficients 9, 18, and 27 all share the factor 9. The smallest power of \(x\) that appears is \(x\), and for \(y\), it is \(y^2\). Therefore, the greatest common factor (GCF) is \(9xy^2\).

Step 2: Factor out the GCF

Once you identify the GCF, factor it out from the entire expression. To do this, divide each term of the polynomial by the GCF \(9xy^2\).- \(\frac{9x^3y^2}{9xy^2} = x^2\)- \(\frac{18x^2y^2}{9xy^2} = 2x\)- \(\frac{27xy^2}{9xy^2} = 3\)So, factoring out \(9xy^2\) from the original expression gives: \(9xy^2(x^2 - 2x + 3)\).

Step 3: Recheck the factorization

Ensure that the expression \(9xy^2(x^2 - 2x + 3)\) is correct by expanding it back. Expand: \(9xy^2(x^2) - 9xy^2(2x) + 9xy^2(3)\) which simplifies to the original expression \(9x^3y^2 - 18x^2y^2 + 27xy^2\). Therefore, the factorization is correct.

Key Concepts

Greatest Common Factor

The greatest common factor (GCF) is a crucial concept in algebra, especially when simplifying expressions. It represents the largest factor that divides all terms in an algebraic expression without leaving a remainder. To find the GCF, start by identifying common factors among the coefficients and variables of each term.
The exercise involves evaluating the expression:

• Coefficients: 9, 18, and 27. Their GCF is 9.
• Variables: The term requiring the smallest power in the expression determines the variable's GCF. Here, the smallest power of \(x\) is \(x\), and for \(y\), it is \(y^2\).
• Thus, the complete GCF of the expression is \(9xy^2\).

Finding the GCF is like discovering the common "building blocks" of the terms, making it easier to simplify the polynomial efficiently.

Polynomials

Polynomials are expressions made up of variables and coefficients, connected by addition, subtraction, or multiplication operations. They are classified by the number of terms and the degree of the largest term. Understanding polynomials is foundational when tackling algebra problems.
In the expression \(9x^3y^2 - 18x^2y^2 + 27xy^2\), this is a polynomial with three terms:

• The first term is \(9x^3y^2\).
• The second term is \(-18x^2y^2\).
• The third term is \(+27xy^2\).

Each term contains a combination of coefficients (9, 18, and 27) and variables raised to different powers. Understanding how to factor polynomials allows one to simplify complex expressions into more manageable parts. This involves recognizing the individual pieces and reassembling them based on common factors.

Algebraic Expressions

Algebraic expressions are a combination of numbers, variables, and operations. They hold the "language" of algebra, allowing mathematicians to describe their findings succinctly and systematically. These expressions can vary in complexity from single-variable operations to multivariable polynomials.
The exercise involves the expression \(9x^3y^2 - 18x^2y^2 + 27xy^2\). This represents a more complex expression due to multiple variables (\(x\) and \(y\)) and terms.

• The goal is to simplify these expressions by finding common elements among the terms and using algebraic rules effectively.
• Simplification involves factoring, where terms are expressed in the form of products.
• By simplifying, one can solve equations more easily or better understand the relationships between variables.

Breaking down complex algebraic expressions makes them more digestible and solvable, thus making the learning and application of algebra enjoyable.