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Chapter 5: Problem 47

Arrange in ascending powers of \(x .\) $$ 4 x+5 x^{3}-x^{6}-9 $$

Short Answer

Expert verified
The polynomial arranged in ascending order of x is -9 + 4x + 5x^3 - x^6

Step by step solution

01

Identify Each Term

First, identify each term in the polynomial: - The term with the constant is -9- The term with the power of 1 is 4x- The term with the power of 3 is 5x^3- The term with the power of 6 is -x^6
02

Arrange Terms by Powers of x

Next, arrange the terms in ascending order of their powers of x. The order of the terms from the lowest power to the highest power is: - -9- 4x- 5x^3- -x^6
03

Write the Final Expression

Write the polynomial with the terms in ascending order of the powers of x: -9 + 4x + 5x^3 - x^6

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Key Concepts

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

Polynomial Terms
In any polynomial, each individual part separated by a plus or minus sign is known as a term. For example, in the polynomial expression given: \(4x + 5x^3 - x^6 - 9\), there are four distinct terms:
  • 4x
  • 5x^3
  • -x^6
  • -9

Each term can consist of:
  • A constant, such as -9.
  • A variable raised to a power, such as \(x^6\).
  • A coefficient, which is the number in front of the variable, such as 5 in \(5x^3\).

Understanding terms is crucial because it helps to identify and manipulate parts of a polynomial with ease. Each term can have a different degree based on the power of its variable.
Ascending Order
Arranging polynomials often requires organizing each term in ascending order of their powers. This means placing the terms from the smallest exponent to the largest. In our example, the powers of the terms are:
  • \(-9\) (which can be seen as \(x^0\)
  • \(4x\) (where the power is 1)
  • \(5x^3\)
  • \(-x^6\)

To arrange these terms in ascending order of the powers of \(x\), start with the term with the lowest power and proceed to the highest: \(-9 + 4x + 5x^3 - x^6\). This ordering makes it easier to read and analyze the polynomial for further operations, such as differentiation or integration.
Powers of x
The power of \(x\) in each term of a polynomial shows the degree of that term. The degree of a term is the exponent of \(x\) in that term. Here are the terms and their corresponding degrees from our example:
  • \(-9\) (degree 0, as there is no \(x\) present)
  • \(4x\) (degree 1)
  • \(5x^3\) (degree 3)
  • \(-x^6\) (degree 6)

The degree of the polynomial itself is the highest degree of any term in the polynomial. In the given problem, since the highest exponent of \(x\) is 6 in \(-x^6\), the polynomial has a degree of 6.
Recognizing the powers is a key aspect when working on polynomial expressions, leading to correct arrangement, simplification, and solving.

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