Problem 6
Question
Factor out the greatest common factor. $$ 6 x^{4} y-4 x^{2} y^{2}+2 x^{2} y^{3} $$
Step-by-Step Solution
Verified Answer
The factored form of the given polynomial is \(2x^2y(3x^2 - 2y + y^2)\).
1Step 1: Identify the GCF of the given polynomial
To find the GCF of the given polynomial, we need to analyze the coefficients and variables of each term. Our polynomial is:
$$
6x^4y - 4x^2y^2 + 2x^2y^3
$$
Coefficients: 6, -4, and 2.
Variables: \(x^4y\), \(x^2y^2\), and \(x^2y^3\).
The GCF of the coefficients is 2 (since 2 is the largest number that evenly divides 6, -4, and 2).
The GCF of the variables is \(x^2y\) (since \(x^2y\) is the largest common expression that can be factored out from \(x^4y\), \(x^2y^2\), and \(x^2y^3\)).
Thus, the GCF of the given polynomial is \(2x^2y\).
2Step 2: Factor out the GCF from each term of the polynomial
We will now factor out the GCF of \(2x^2y\) from each term of the polynomial:
$$
6x^4y - 4x^2y^2 + 2x^2y^3 = 2x^2y(3x^2 - 2y + y^2)
$$
In each term, we have divided the term by the GCF (2, \(x^2\), and y respectively) and then multiplied the GCF outside the parentheses.
3Step 3: Check the factored expression
Check if the factored expression is correct by multiplying the GCF with the terms inside the parentheses:
$$
2x^2y(3x^2 - 2y + y^2) = 6x^4y - 4x^2y^2 + 2x^2y^3
$$
The factored expression is correct since it matches the original polynomial after expanding.
That's it! The factored form of the given polynomial is:
$$
2x^2y(3x^2 - 2y + y^2)
$$
Key Concepts
Polynomial FactoringVariables and CoefficientsFactored Expressions
Polynomial Factoring
Polynomial factoring is a mathematical process used to simplify polynomials by expressing them in terms of their simplest building blocks, known as factors. Imagine breaking down a complex concept into its most basic elements to better understand it. This is essentially what we do with polynomial factoring.
To factor a polynomial, we look for common attributes shared by all terms, such as numerical coefficients or algebraic variables, which we can simplify. By identifying and extracting these common factors, we can express the polynomial in a more concise way.
In our example, we identified the greatest common factor of all terms in the polynomial \(6x^4y - 4x^2y^2 + 2x^2y^3\), allowing us to rewrite it more neatly as \(2x^2y(3x^2 - 2y + y^2)\).
Having factored expressions is not just about simplification, but also about solving equations more efficiently and gaining deeper insights into the structure of the polynomial.
To factor a polynomial, we look for common attributes shared by all terms, such as numerical coefficients or algebraic variables, which we can simplify. By identifying and extracting these common factors, we can express the polynomial in a more concise way.
In our example, we identified the greatest common factor of all terms in the polynomial \(6x^4y - 4x^2y^2 + 2x^2y^3\), allowing us to rewrite it more neatly as \(2x^2y(3x^2 - 2y + y^2)\).
Having factored expressions is not just about simplification, but also about solving equations more efficiently and gaining deeper insights into the structure of the polynomial.
Variables and Coefficients
Understanding the variables and coefficients in a polynomial is critical for successful factoring. A polynomial is built from terms, each consisting of a coefficient and variables raised to exponent powers.
Coefficients are the numerical parts. They tell us how many times the variable terms are being summed. For instance, in the polynomial \(6x^4y - 4x^2y^2 + 2x^2y^3\), the coefficients are 6, -4, and 2. When factoring, we emphasize finding the greatest number that divides each coefficient evenly, as it will become part of the greatest common factor.
Variables, on the other hand, represent the unknowns. They can appear in various forms depending on the powers they are raised to, like \(x^4\) or \(y^2\). In our problem, both variables \(x\) and \(y\) are essential, with their powers indicating their multiplicity in the factors. Identifying the smallest power common to all terms for each variable helps us determine what can be factored out. Here, \(x^2y\) serves this purpose, acting as part of the greatest common factor.
Coefficients are the numerical parts. They tell us how many times the variable terms are being summed. For instance, in the polynomial \(6x^4y - 4x^2y^2 + 2x^2y^3\), the coefficients are 6, -4, and 2. When factoring, we emphasize finding the greatest number that divides each coefficient evenly, as it will become part of the greatest common factor.
Variables, on the other hand, represent the unknowns. They can appear in various forms depending on the powers they are raised to, like \(x^4\) or \(y^2\). In our problem, both variables \(x\) and \(y\) are essential, with their powers indicating their multiplicity in the factors. Identifying the smallest power common to all terms for each variable helps us determine what can be factored out. Here, \(x^2y\) serves this purpose, acting as part of the greatest common factor.
Factored Expressions
A factored expression is the result of breaking down a polynomial into its simplest form using the greatest common factor and other techniques. Factored expressions are crucial because they simplify complex polynomials, making them easier to work with and understand.
In our example, we used polynomial factoring to derive: \(2x^2y(3x^2 - 2y + y^2)\). This expression is cleaner and more manageable. The components inside the parentheses, such as \(3x^2\), \(-2y\), and \(y^2\), reflect what remains after factoring out the GCF \(2x^2y\).
Factored expressions are especially useful in solving equations, as they can reveal roots more comprehensively. Additionally, they play a critical role in different mathematical applications, such as simplifying rational expressions or finding zero points in polynomial graphs. By mastering the skill of factoring, you can tackle more advanced mathematical challenges with confidence.
In our example, we used polynomial factoring to derive: \(2x^2y(3x^2 - 2y + y^2)\). This expression is cleaner and more manageable. The components inside the parentheses, such as \(3x^2\), \(-2y\), and \(y^2\), reflect what remains after factoring out the GCF \(2x^2y\).
Factored expressions are especially useful in solving equations, as they can reveal roots more comprehensively. Additionally, they play a critical role in different mathematical applications, such as simplifying rational expressions or finding zero points in polynomial graphs. By mastering the skill of factoring, you can tackle more advanced mathematical challenges with confidence.
Other exercises in this chapter
Problem 6
Solve the given equation. $$ 2-3 y=8 $$
View solution Problem 6
Rewrite the number without using exponents. $$ -4^{2} $$
View solution Problem 6
Classify the number as to type. (For example, \(\frac{1}{2}\) is rational and real, whereas \(\sqrt{5}\) is irrational and real.) $$ -\sqrt{5} $$
View solution Problem 6
Evaluate the expression. $$ -\left(-\frac{4}{5}\right)^{3} $$
View solution