/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 98 Factor completely: \(2 x^{6}+20 ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 98

Factor completely: \(2 x^{6}+20 x^{5} y+50 x^{4} y^{2}\) (Section 6.5, Example 8)

Short Answer

Expert verified
The completely factored form of the polynomial \(2 x^{6}+20 x^{5} y+50 x^{4} y^{2}\) is \(2x^{4}(x+5y)^{2}\)

Step by step solution

01

Identify the Greatest Common Factor (GCF)

First observe all the terms in the given expression: \(2 x^{6}, 20 x^{5} y, 50 x^{4} y^{2}\). The GCF is the product of the lowest power of \(x\) and the common numerical factor. From the numerical coefficients 2, 20, and 50, the common factor is 2. Among the terms, the lowest power of \(x\) is \(x^{4}\), and no term has \(y\) as a common factor. Therefore, the GCF is \(2x^{4}\).
02

Factor out the GCF

Factor out the GCF \(2x^{4}\) from each term in the polynomial. This means that each term in the polynomial will be divided by \(2x^{4}\). Thus the expression becomes: \[2x^{4}(x^{2}+10xy+25y^{2})\]
03

Verify if the Inner Expression Can Be Factored Further

The inner expression \(x^{2}+10xy+25y^{2}\) can be factored further as it is in the form of a perfect square trinomial (\(a^{2}+2ab+b^{2}\)). This can be written as \((a+b)^{2}\). In the inner expression, \(a = x\), \(b = 5y\). Therefore, the inner expression can be further factored as \((x+5y)^{2}\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Greatest Common Factor (GCF)
When dealing with polynomials, the first step to simplify or factor them completely is often to find the Greatest Common Factor (GCF). The GCF is the largest factor that divides all terms in a polynomial evenly. It simplifies expressions by reducing each term significantly, making it easier to work with.

To find the GCF of a polynomial expression, follow these steps:
  • Check each term of the polynomial.
  • Identify the numerical factors shared by all coefficients. For example, with coefficients 2, 20, and 50, the GCF is 2, as it is the largest number that divides all.
  • Look at the variables and pick the lowest exponent for those common variables across all terms. In this case, for the variable x, this is \(x^4\).
After identifying the GCF, factor it out from the original polynomial. This reduces the complexity of the expression and is the foundation for further factoring steps.
Perfect Square Trinomial
A perfect square trinomial is a special kind of trinomial that can be written as the square of a binomial. This occurs when a trinomial fits the pattern \(a^2 + 2ab + b^2\), which can be factored into \((a+b)^2\). The recognition of a perfect square trinomial is critical when simplifying polynomial expressions.

To spot a perfect square trinomial:
  • Identify if the first and last terms are perfect squares. Here, \(x^2\) and \(25y^2\) are perfect squares.
  • Check if the middle term is twice the product of the roots of the first and last terms. For instance, in \(10xy\), this is indeed \(2 imes x imes 5y\).
Once confirmed as a perfect square, rewrite the trinomial \(x^2 + 10xy + 25y^2\) as \((x+5y)^2\). This process enables easier manipulation and solution of the expression in real mathematical applications.
Polynomial Expressions
Polynomial expressions are composed of terms, which themselves contain variable bases with integer exponents, and coefficients. Understanding polynomials, and how to manipulate them, is foundational in algebra. It allows solving complex problems by breaking them down into simpler parts.

Key features of polynomial expressions include:
  • Each term is a product of a constant and a variable raised to an integer power.
  • The degree of the polynomial is determined by the highest exponent of the variable present.
  • They can often be factored or expanded for simplification.
Factoring polynomials, like the given expression \(2x^6 + 20x^5y + 50x^4y^2\), requires identifying patterns such as common factors and special trinomials. Mastery in this area is achieved through practice and understanding of fundamental properties, making algebra powerful and intuitive.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. It's not a good thing for a business if \(R(x)>C(x)\)

Solve each inequality using a graphing utility. Graph each of the three parts of the inequality separately in the same viewing rectangle. The solution set consists of all values of \(x\) for which the graph of the linear function in the middle lies between the graphs of the constant functions on the left and the right. $$-1<\frac{x+4}{2}<3$$

The toll to a bridge is \(\$ 3.00 .\) A three-month pass costs \(\$ 7.50\) and reduces the toll to \(\$ 0.50 .\) A six-month pass costs \(\$ 30\) and permits crossing the bridge for no additional fee. How many crossing per three-month period does it take for the three-month pass to be the best deal?

A basic cellphone plan costs \(\$ 20\) per month for 60 calling minutes. Additional time costs \(\$ 0.40\) per minute. The formula $$C=20+0.40(x-60)$$ gives the monthly cost for this plan, \(C\), for \(x\) calling minutes, where \(x>60 .\) How many calling minutes are possible for a monthly cost of at least \(\$ 28\) and at most \(\$ 40 ?\)

Solve each inequality using a graphing utility. Graph each of the three parts of the inequality separately in the same viewing rectangle. The solution set consists of all values of \(x\) for which the graph of the linear function in the middle lies between the graphs of the constant functions on the left and the right. $$2 \leq 4-x \leq 7$$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.