Problem 48 - Algebra 2

Question

Factor completely. If the polynomial is not factorable, write prime. $$ 7 x y^{3}-14 x^{2} y^{5}+28 x^{3} y^{2} $$

Step-by-Step Solution

Answer

The polynomial factors as \(7xy^2(y - 2xy^3 + 4x^2)\).

Step 1: Identify Common Factors

First, examine each term in the polynomial to identify any common factors. The terms given are: \(7xy^3\), \(-14x^2y^5\), and \(28x^3y^2\). The common numerical factor is 7, as it divides each of these coefficients. Additionally, the lowest power of \(x\) present in all terms is \(x^1\), and the lowest power of \(y\) present is \(y^2\). Therefore, the greatest common factor (GCF) is \(7xy^2\).

Step 2: Factor Out the GCF

Factor out the greatest common factor \(7xy^2\) from each term in the polynomial. Divide each of the terms by \(7xy^2\):\[\begin{align*}7xy^3 & : \quad (7xy^3) \div (7xy^2) = y,\-14x^2y^5 & : \quad (-14x^2y^5) \div (7xy^2) = -2xy^3,\28x^3y^2 & : \quad (28x^3y^2) \div (7xy^2) = 4x^2.\end{align*}\]Thus, the polynomial is:\[7xy^2(y - 2xy^3 + 4x^2).\]

Step 3: Simplify and Verify

Check if the remaining polynomial \(y - 2xy^3 + 4x^2\) can be further factored. However, there are no additional common factors or recognizable patterns (such as quadratic form or difference of squares) in this polynomial. Therefore, it is already in its simplest factored form.

Key Concepts

Greatest Common Factor

The first step in solving polynomial factorization problems is to identify the Greatest Common Factor (GCF). The GCF is the largest factor that each term in the polynomial shares. This helps us to simplify and break down more complex expressions.
In the given polynomial, which is composed of the terms: \(7xy^3\), \(-14x^2y^5\), and \(28x^3y^2\), we look for common factors.

• The numerical coefficients are 7, -14, and 28. The greatest common divisor of these numbers is 7.
• For the variables, we identify the smallest power of \(x\) in all the terms as \(x^1\). This is our common \(x\) factor.
• Likewise, for \(y\), \(y^2\) is the smallest power present in all terms. Thus, \(y^2\) is the common \(y\) factor.

Combining these, the GCF of the polynomial is \(7xy^2\). Factoring the GCF out of the polynomial is crucial as it simplifies the remaining expression and prepares it for further factorization if possible.

Monomial

A monomial is simply a single term in a polynomial. It can be a single number, a variable, or numbers and variables multiplied together.
In the exercise, each part of the polynomial \(7xy^3\), \(-14x^2y^5\), and \(28x^3y^2\) is a monomial. They are:

• First Monomial: \(7xy^3\)
• Second Monomial: \(-14x^2y^5\)
• Third Monomial: \(28x^3y^2\)

Each monomial consists of a coefficient (7, -14, 28) and variables raised to integer powers \(x^a y^b\). Recognizing monomials helps in simplifying the polynomial and deciding the best approach to factor them.It is especially useful to understand monomials when determining the Greatest Common Factor.

Factoring by Grouping

Factoring by grouping is a method used to simplify a polynomial further. Usually, this is applied when you have four or more terms and involves pairing the terms to find common factors, but it can also help in recognizing patterns.
For the given exercise, after factoring out the GCF \(7xy^2\) from the polynomial, we are left with \( (y - 2xy^3 + 4x^2)\).
At this stage, it's important to check if there’s an opportunity to group these terms into smaller parts that can be factored further. However, in this specific instance, the terms do not lend themselves naturally to additional grouping or factorization due to the lack of common patterns or factors beyond the GCF already extracted.
Thus, the factorization by grouping doesn't apply further here because the remaining polynomial is in its simplest form.

Simplification Steps

Simplification is a crucial step in mathematics that involves reducing expressions to their simplest form. In this context, it means factoring out the Greatest Common Factor and verifying if the remaining polynomial can be further reduced.
After extracting the GCF \(7xy^2\), you are left with \( (y - 2xy^3 + 4x^2)\). It's essential to:

• Check each term in the simplified polynomial to see if a common factor can once again be extracted.
• Look for special patterns, such as quadratics or the difference of squares, which could allow further factorization.

In this scenario, the polynomial \( (y - 2xy^3 + 4x^2)\) does not simplify further because no additional factoring patterns are present. Hence, it remains in its reduced form after the initial extraction of the GCF.