Problem 40 - Algebra for College Students

Question

Factor completely. $$8 x^{5} y^{3}-6 x^{4} y^{5}+12 x^{2} y^{3}$$

Step-by-Step Solution

Answer

The expression factors to \(2x^2y^3(4x^3 - 3x^2y^2 + 6)\).

Step 1: Identify Common Factors

Look for the greatest common factor (GCF) present in all terms of the expression. All the terms have common factors of 2, at least two x's, and at least three y's. The GCF is therefore \(2x^2y^3\).

Step 2: Apply the Greatest Common Factor

Divide each term in the expression by the GCF, \(2x^2y^3\). This gives: \[\frac{8x^5y^3}{2x^2y^3} = 4x^3\]\[\frac{-6x^4y^5}{2x^2y^3} = -3x^2y^2\]\[\frac{12x^2y^3}{2x^2y^3} = 6\]Thus, the expression becomes \(2x^2y^3(4x^3 - 3x^2y^2 + 6)\).

Step 3: Further Factor if Possible

Check if \(4x^3 - 3x^2y^2 + 6\) can be factored further. After examination, it shows no further factorization is possible beyond the linear terms already factored out.

Key Concepts

Greatest Common Factor (GCF)

The concept of the Greatest Common Factor (GCF) is fundamental in simplifying algebraic expressions. The GCF is the largest factor that divides each term in an expression without leaving a remainder. It helps in breaking down complex expressions into simpler ones. To find the GCF, we need to look at both the numerical coefficients and the variables involved.

• For numbers, identify the largest integer that divides all the coefficients.
• For variables, consider the lowest power of each variable present in every term.

By factoring out the GCF, you simplify the expression and make further algebraic operations more manageable.
The GCF is particularly useful in polynomial division. It can break down polynomials into products of simpler expressions, offering a clearer view of the polynomial's structure.

Polynomials

Polynomials are expressions that consist of variables and coefficients. They are combined using addition, subtraction, and multiplication. Each term in a polynomial is a product of a number (coefficient) and a variable raised to a power.

• The degree of a polynomial is determined by the highest power of the variable present in the expression.
• Polynomials can be one or more terms long. Examples are a single term, like \(5x^2\) (a monomial), or multiple terms, like \(2x^2 + 3x + 4\) (a trinomial).

Understanding polynomials is crucial since they are a core part of algebra. They form the basis for solving complex equations and inequalities.
Recognizing the form and components of polynomials helps in their manipulation and simplification. This knowledge is foundational when factoring polynomials or solving polynomial equations.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions using established rules. This process can include tasks such as factoring, expanding, and solving equations. Factoring is especially useful to simplify expressions or solve equations. It involves identifying factors within an expression, like using the GCF or recognizing special forms such as the difference of squares.

• Algebraic manipulation aims to express complex algebraic forms in simpler, more manageable ones.
• Identifying when and how to factor or expand expressions is crucial for solving problems efficiently.

Furthermore, algebraic manipulation often helps us to see the relationships between different expressions clearly, offering insights into solving them. It is a vital skill in algebra that applies to a wide range of problems, from basic to advanced levels.