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Chapter 5: Problem 50

Arrange in ascending powers of \(x .\) $$ 2 a x-9 a b+4 x^{5}-7 b x^{2} $$

Short Answer

Expert verified
\(-9ab, 2ax, -7bx^{2}, 4x^{5}\)

Step by step solution

01

Identify the terms

First, identify all the terms in the given expression: \(2ax, -9ab, 4x^{5}, -7bx^{2}\)
02

Determine the powers of \(x\) for each term

Next, find the power of \(x\) in each term: \(2ax\) has \(x\) raised to the power of 1, \(-9ab\) has no \(x\) (considered \(x^0\)), \(4x^{5}\) has \(x\) raised to the power of 5, \(-7bx^{2}\) has \(x\) raised to the power of 2.
03

Arrange the terms in ascending order of the powers of \(x\)

Now, write the terms in ascending order of the powers of \(x\): \( -9ab, 2ax, -7bx^{2}, 4x^{5}\)

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Key Concepts

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

polynomial terms
Polynomials are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. Each combination of variables and their powers, along with their respective coefficients, is referred to as a term. For instance, in the expression given:
  • The term 2ax involves the variable x and has a coefficient of 2a.
  • The term -9ab does not involve x at all.
  • 4x^5 includes x raised to the 5th power and a coefficient of 4.
  • -7bx^2 involves x raised to the 2nd power and a coefficient -7b.
Each term's variable and power combination makes up the structure of the polynomial, and understanding this is crucial to manipulating and ordering polynomial expressions correctly.
ascending order
Arranging terms in ascending order means sorting them from the smallest power to the largest power of a particular variable. For example, in the exercise:
  • You identify the power of x in each term.
  • Then, you start with the term that has the smallest power of x.
  • Finally, you arrange each subsequent term according to the increasing power of i(x).
If a term does not have the variable, it is considered as having the power of 0 . In the given exercise, -9ab comes first because it has no (i)x, making the power of xi(either (i)x^0 - which is 1). Following that, 2ax comes next as x is raised once (x^1) .Then -7bx^2 is picked as xi(x^2). Finally,
  • 4x^5 is arranged last as it is raised to the 5th power (i(x^5).
    • powers of x
    The power of a variable in a polynomial term tells us how many times that variable is multiplied by itself. For any variable x, (i(x^n) signifies that x is multiplied by itself n times. Here's the breakdown for each term in the given exercise:
    • 2axi(x^1) signifies x to the power of 1.
    • -9ab has no x (or x^0, which is 1), so its power is 0.
    • 4x^5 clearly shows that xileo raised to the power of 5.
    • -7bx^2 indicates x is raised to the power of 2.

    Recognizing these powers is essential when it comes to arranging terms, simplifying polynomials, and performing arithmetic operations like addition, subtraction, multiplication, and division involving polynomials. By effectively leveraging the powers of a variable, you can gain a deeper understanding of the polynomial's structure and behavior.

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