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Chapter 5: Problem 34

Write the following polynomials in standard form. $$ a 3-a 9+6 a 5-a+3-a 4 $$

Short Answer

Expert verified
-a^9 + 6a^5 - a^4 + a^3 - a + 3

Step by step solution

01

Identify Like Terms

Start by identifying and grouping like terms in the polynomial. The given polynomial is not in order:\[ a^3 - a^9 + 6a^5 - a + 3 - a^4 \]Order the terms by their exponents such as \(a^9, a^5, a^4, a^3, a^1\) and constant.
02

Rearrange Terms by Exponents

Rearrange the terms from highest exponent to lowest:\[ -a^9 + 6a^5 - a^4 + a^3 - a + 3 \]
03

Write the Polynomial in Standard Form

The polynomial is now arranged from the highest power to the lowest power, which is the standard form:\[ -a^9 + 6a^5 - a^4 + a^3 - a + 3 \]

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Key Concepts

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

Standard Form
When dealing with polynomials, putting them in standard form means arranging them so that the terms are ordered by their exponents from highest to lowest. This helps to simplify the polynomial and makes it easier to work with. For example, if you have a polynomial such as \(-a^9 + 6a^5 - a^4 + a^3 - a + 3\), you should rearrange it starting with the highest exponent, which is \(-a^9\).

In standard form:
  • The term with the largest exponent comes first.
  • Followed by the next largest exponent, and so on.
  • Finally ending with the constant term, if there is one.
You can think of standard form as a way to "tidy up" your polynomial, making sure everything is in its correct order. This helps in operations like addition, subtraction, and finding derivatives or integrals.
Exponents
Exponents are a way to represent repeated multiplication. In the context of polynomials, each term consists of a variable raised to a certain power, known as the exponent. For instance, in the term \(6a^5\), the exponent is 5, which means the variable \(a\) is multiplied by itself 5 times.

Understanding exponents is key in arranging polynomials and performing operations such as multiplication and division.
  • The exponent tells you how many times to multiply the base by itself.
  • A larger exponent means the term has more influence on the polynomial's graph as the variable increases or decreases.
  • An exponent of 0 means the term is equal to 1, since any number raised to the power of 0 is 1.
By mastering exponents, you can manipulate and simplify polynomials more effectively.
Like Terms
Like terms in polynomials are terms that have the exact same variable part. This means they can be combined together by simply adding or subtracting their coefficients. For example, \(3a^3\) and \(-5a^3\) are like terms because they both have \(a^3\) as their variable component.

To work effectively with polynomials:
  • Identify like terms so you can combine them to simplify the polynomial.
  • Ensure that only the coefficients in front of the like terms are added or subtracted.
  • Non-like terms (with different exponents) should remain separate.
Finding and combining like terms helps to simplify polynomials, making them easier to understand and solve. Recognizing these terms is crucial for writing and working with polynomials in standard form.

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