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Chapter 3: Problem 27

Subtract and write the resulting polynomial in descending order of degree. $$(5 x+9)-(3 x+2)$$

Short Answer

Expert verified
Question: Subtract the given polynomials: (5x + 9) - (3x + 2). Answer: 2x + 7

Step by step solution

01

Distribute the negative sign

Make sure to distribute the negative sign from the subtraction operation (-) to every term inside the parentheses. $$(5x + 9) - (3x + 2) = 5x + 9 - 3x - 2$$
02

Combine like terms

Combine the terms with the same variables raised to the same power (like terms): $$5x - 3x + 9 - 2$$
03

Simplify the expression

Simplify the expression further by calculating the result of the addition and subtraction: $$2x + 7$$
04

Write the resulting polynomial in descending order of degree

The polynomial is already in descending order of degree, so the final answer is: $$2x + 7$$

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Key Concepts

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

Understanding Like Terms
When dealing with polynomials, it's crucial to understand the concept of like terms. Like terms are terms that have the same variables raised to the same powers. This means they can be added or subtracted from one another. For example, in the expression \(5x - 3x + 9 - 2\), both \(5x\) and \(-3x\) are like terms because they both contain the variable \(x\) and are raised to the power of 1.

When you combine like terms, you essentially add or subtract their coefficients (the numbers in front of the variables). In our example, combining \(5x\) and \(-3x\) results in \(2x\).
  • Identify the like terms.
  • Add or subtract their coefficients.
Mastering this skill simplifies polynomials and is essential for solving algebraic expressions effectively.
Arranging in Descending Order
Once you have simplified a polynomial, the next step is to ensure it's arranged in descending order of degree. The degree of a term is determined by the exponent of its variable. For instance, in the term \(2x^2\), the degree is 2.

Arranging in descending order means you list the terms starting with the highest degree followed by lower degrees. In our final expression, \(2x + 7\), \(2x\) comes first as it has a degree of 1, compared to \(7\), which is technically \(7x^0\) and has a degree of 0.
  • Identify the degree of each term.
  • Rearrange terms from highest to lowest degree.
This organization makes it easier to evaluate and compare polynomials.
Simplifying Expressions
Simplifying expressions involves combining like terms and performing any possible arithmetic operations. During simplification, it's important to follow the order of operations carefully.

In the problem \((5x + 9) - (3x + 2)\), we started by distributing the negative sign, which allowed us to remove the parentheses. This gave us \(5x + 9 - 3x - 2\). By combining the like terms \(5x\) and \(-3x\), we achieved \(2x\). Then, we simplified the constants \(9 - 2\) to get \(7\).
  • Distribute any negative signs.
  • Combine like terms.
  • Perform basic arithmetic operations.
Finishing these steps gives the polynomial in its simplest form, making it easier for further calculations or analyses.

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