Thank you for visiting Determine whether the polynomial tex 25 y 2 y 3 25 tex is completely factored If not factor the polynomial A Completely factored B tex. This page is designed to guide you through key points and clear explanations related to the topic at hand. We aim to make your learning experience smooth, insightful, and informative. Dive in and discover the answers you're looking for!
Answer :
To determine whether the polynomial [tex]\(25 y^2(y^3 - 25)\)[/tex] is completely factored, let's follow these steps:
1. Identify the Structure of the Polynomial:
The polynomial is given as [tex]\(25 y^2(y^3 - 25)\)[/tex]. It's structured as a product of [tex]\(25 y^2\)[/tex] and [tex]\((y^3 - 25)\)[/tex].
2. Check if Each Part is Factored Completely:
- The term [tex]\(25 y^2\)[/tex] is already in its simplest factored form, as it consists of a constant multiplier [tex]\(25\)[/tex] and a simple power of [tex]\(y^2\)[/tex].
- Next, look at [tex]\((y^3 - 25)\)[/tex]. This expression might be a candidate for further factoring. Notice that it resembles a difference of cubes since [tex]\(25 = 5^2\)[/tex].
3. Factor the Difference of Cubes (if applicable):
A difference of cubes follows the formula:
[tex]\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\][/tex]
Here, [tex]\(y^3\)[/tex] can be seen as [tex]\(a^3\)[/tex] with [tex]\(a = y\)[/tex], but [tex]\(25\)[/tex] is not a perfect cube because it cannot be expressed in the form [tex]\(b^3\)[/tex].
Since [tex]\(25\)[/tex] is not a cube, [tex]\(y^3 - 25\)[/tex] is not a difference of cubes and does not factor further using this method.
4. Conclude:
Since [tex]\(y^3 - 25\)[/tex] does not factor further with integer coefficients and no standard method could simplify it further, the polynomial [tex]\(25 y^2(y^3 - 25)\)[/tex] is already in its completely factored form.
Thus, the polynomial [tex]\(25 y^2(y^3 - 25)\)[/tex] is indeed completely factored.
1. Identify the Structure of the Polynomial:
The polynomial is given as [tex]\(25 y^2(y^3 - 25)\)[/tex]. It's structured as a product of [tex]\(25 y^2\)[/tex] and [tex]\((y^3 - 25)\)[/tex].
2. Check if Each Part is Factored Completely:
- The term [tex]\(25 y^2\)[/tex] is already in its simplest factored form, as it consists of a constant multiplier [tex]\(25\)[/tex] and a simple power of [tex]\(y^2\)[/tex].
- Next, look at [tex]\((y^3 - 25)\)[/tex]. This expression might be a candidate for further factoring. Notice that it resembles a difference of cubes since [tex]\(25 = 5^2\)[/tex].
3. Factor the Difference of Cubes (if applicable):
A difference of cubes follows the formula:
[tex]\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\][/tex]
Here, [tex]\(y^3\)[/tex] can be seen as [tex]\(a^3\)[/tex] with [tex]\(a = y\)[/tex], but [tex]\(25\)[/tex] is not a perfect cube because it cannot be expressed in the form [tex]\(b^3\)[/tex].
Since [tex]\(25\)[/tex] is not a cube, [tex]\(y^3 - 25\)[/tex] is not a difference of cubes and does not factor further using this method.
4. Conclude:
Since [tex]\(y^3 - 25\)[/tex] does not factor further with integer coefficients and no standard method could simplify it further, the polynomial [tex]\(25 y^2(y^3 - 25)\)[/tex] is already in its completely factored form.
Thus, the polynomial [tex]\(25 y^2(y^3 - 25)\)[/tex] is indeed completely factored.
Thank you for reading the article Determine whether the polynomial tex 25 y 2 y 3 25 tex is completely factored If not factor the polynomial A Completely factored B tex. We hope the information provided is useful and helps you understand this topic better. Feel free to explore more helpful content on our website!
- You are operating a recreational vessel less than 39 4 feet long on federally controlled waters Which of the following is a legal sound device
- Which step should a food worker complete to prevent cross contact when preparing and serving an allergen free meal A Clean and sanitize all surfaces
- For one month Siera calculated her hometown s average high temperature in degrees Fahrenheit She wants to convert that temperature from degrees Fahrenheit to degrees
Rewritten by : Jeany
