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Chapter 5: Problem 49

Arrange in ascending powers of \(x .\) $$ 2 x^{2} y+5 x y^{3}-x^{3}+8 y $$

Short Answer

Expert verified
The expression in ascending powers of \(x\) is: \[8y + 5xy^{3} + 2x^{2}y - x^{3}\]

Step by step solution

01

Identify the terms

The expression consists of four terms: \(2x^{2}y\), \(5xy^{3}\), \(-x^{3}\), and \(8y\).
02

Determine the powers of x

The powers of \(x\) in each term are as follows:- \(2x^{2}y\) has a power of 2 for \(x\)- \(5xy^{3}\) has a power of 1 for \(x\)- \(-x^{3}\) has a power of 3 for \(x\)- \(8y\) has a power of 0 for \(x\)
03

Arrange terms by ascending powers of x

Rearrange the terms based on their powers of \(x\) (from lowest to highest):1. \(8y\) (\(x\) power: 0)2. \(5xy^{3}\) (\(x\) power: 1)3. \(2x^{2}y\) (\(x\) power: 2)4. \(-x^{3}\) (\(x\) power: 3)
04

Write the final expression

The rearranged expression in ascending powers of \(x\) is:\[8y + 5xy^{3} + 2x^{2}y - x^{3}\]

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Key Concepts

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

Ascending Powers
When arranging polynomial terms, one common approach is to order them by the ascending powers of a variable, such as x. Ascending powers means that the exponents of the variable increase as you move from one term to the next.

For instance, in the given exercise, we start by identifying the exponents of x in each term:
  • \(2x^2y\) has an exponent of 2.
  • \(5xy^3\) has an exponent of 1.
  • \(-x^3\) has an exponent of 3.
  • \(8y\) has an exponent of 0 (since there is no \(x\) term in \(8y\)).


By ordering these terms from the lowest to the highest power of \(x\), we get:
  • \(8y\) (x power: 0),
  • \(5xy^3\) (x power: 1),
  • \(2x^2y\) (x power: 2),
  • \(-x^3\) (x power: 3).
Polynomial Terms
A polynomial is an algebraic expression made up of terms that are added or subtracted. Each term consists of coefficients and variables raised to an exponent.

In our exercise, the polynomial is \(2x^2y + 5xy^3 - x^3 + 8y\). Here鈥檚 how the terms break down:
  • \(2x^2y\) - This term has a coefficient of 2, \(x\) raised to the power of 2, and \(y\).
  • \(5xy^3\) - Here, the coefficient is 5, \(x\) raised to the power of 1, and \(y\) raised to the power of 3.
  • \(-x^3\) - The coefficient here is -1, and \(x\) is raised to the power of 3.
  • \(8y\) - This term has a coefficient of 8 and \(y\) with no \(x\) term, implying \(x^0\) where the power of \(x\) is 0.


Recognizing and understanding each term is essential for arranging and performing operations with polynomials.
Algebraic Expression
An algebraic expression is a combination of numbers, variables, and operations (such as addition and subtraction). Unlike an equation, it does not have an equal sign.

The expression in our exercise, \(2x^2y + 5xy^3 - x^3 + 8y\), is an algebraic expression because it combines different terms using addition and subtraction.

Each term in an algebraic expression can be separated and rearranged, particularly when guided by conditions like arranging by ascending powers of a variable.

Here are some key points to remember:
  • Identify each term and its components (coefficients and variables).
  • Understand the exponents of each variable in the terms.
  • Arrange the terms if needed, based on given criteria like ascending powers.


By mastering how to manage and manipulate algebraic expressions, you can solve more complex algebra problems with confidence.

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