Concept Check When directed to completely factor the polynomial 4x^2y^5-8xy^3,a student wrote 2xy^3(2xy^2-4). When the teacher did not give him full credit, he complained because when his answer is multiplied out, the result is the original polynomial. Give the correct answer.

Verified step by step guidance

1. Step 1: Identify the greatest common factor (GCF) of the terms in the polynomial.
2. Step 2: The terms in the polynomial are 4x^2y^5 and -8xy^3. Find the GCF of the coefficients (4 and 8) and the variables (x^2 and x, y^5 and y^3).
3. Step 3: The GCF of the coefficients 4 and 8 is 4. The GCF of x^2 and x is x. The GCF of y^5 and y^3 is y^3. Therefore, the overall GCF is 4xy^3.
4. Step 4: Factor out the GCF from each term in the polynomial.
5. Step 5: Write the polynomial as 4xy^3(____), filling in the blank with the terms obtained after factoring out the GCF from each original term.

Key Concepts

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its simpler components, or factors. This process is essential for simplifying expressions and solving equations. A polynomial is fully factored when it cannot be expressed as a product of polynomials of lower degree. Understanding how to identify common factors and apply techniques like grouping or using the distributive property is crucial for accurate factoring.

Greatest Common Factor (GCF)

The Greatest Common Factor (GCF) is the largest factor that divides two or more numbers or terms without leaving a remainder. In polynomial factoring, identifying the GCF allows for the simplification of the polynomial by factoring it out. For the polynomial 4x^2y^5 - 8xy^3, the GCF is 4xy^3, which should be factored out first to simplify the expression correctly.

Correctness in Factoring

Correctness in factoring means that the factored form must accurately represent the original polynomial. While a student may arrive at a product that equals the original polynomial, the factored form must be fully simplified and include all factors. In this case, the student's factorization was incomplete, as it did not include the GCF, leading to an incorrect representation of the polynomial's factors.