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Chapter 3: Problem 1

Add and write the resulting polynomial in descending order of degree. $$(2 x+5)+(3 x+1)$$

Short Answer

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Question: Add the polynomial expressions (2x + 5) and (3x + 1) and write the resulting polynomial in descending order of degree. Answer: The sum of the given polynomial expressions is 5x + 6.

Step by step solution

01

Identify Like Terms

In the given polynomials \((2x + 5)\) and \((3x + 1)\), the like terms are the ones that have the same variable with the same exponent: \(2x\) and \(3x\), and the constants \(5\) and \(1\).
02

Add the Coefficients of the Like Terms

Now, we need to add the coefficients of the like terms. For the \(x\) terms, we have \(2x + 3x\). To add them, just add their coefficients: \(2 + 3 = 5\). So the sum of these terms is \(5x\). For the constant terms, we have \(5 + 1 = 6\). So the sum of the constants is \(6\).
03

Write the Resulting Polynomial

Now that we have found the sum of the like terms, we need to write the resulting polynomial. Since we have \(5x\) and \(6\) as the sums of the like terms, the resulting polynomial is \(5x + 6\).
04

Write the Polynomial in Descending Order of Degree

The resulting polynomial, \(5x + 6\), is already in descending order of degree. The \(x\) term has a degree of 1, and the constant term has a degree of 0. So, the final answer is: $$ (2x + 5) + (3x + 1) = 5x + 6$$

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Key Concepts

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

Like Terms
When adding polynomials, it's crucial to identify "like terms". Like terms are components of expressions that share the same variable and exponent. For example, in the polynomials
  • (2x + 5) and (3x + 1),
we notice that the terms with the variable "\(x\)"鈥攏amely, \(2x\) and \(3x\)鈥攁re like terms. This is because they both have the variable \(x\) raised to the same power, which is 1.
Unlike terms, such as \(2x\) and the constant 5, do not match in both variable and exponent and therefore cannot be directly combined.Like terms are easy to handle when it comes to adding polynomials, as they simplify the process by allowing us to combine their coefficients, making the calculation straightforward.
Coefficients
In any polynomial, numbers preceding the variables are known as coefficients. They play an essential role during addition or subtraction of like terms.
So, if you see terms like \(2x\) and \(3x\), the coefficients are 2 and 3, respectively.
While combining like terms:
  • Add the coefficients to create a new term with the same variable part.
  • For example, adding \(2x\) to \(3x\) means adding their coefficients: \(2 + 3 = 5\).
In the combined expression, this results in \(5x\). Pay attention as coefficients will always be added or subtracted as per the operation given in the exercise.
Descending Order of Degree
Once we have combined the like terms in a polynomial addition problem, the expression should be simplified further by arranging the terms in descending order of degree. This means we start with the term having the highest exponent and proceed to the one with the lowest.
For instance, after you combine \(2x + 3x = 5x\) and the constants \(5 + 1 = 6\), the resulting expression is \(5x + 6\).
  • Here, \(5x\) is placed first because it is of degree 1.
  • The constant 6, having a degree of 0, follows.
This results in a polynomial that is easy to understand and more organized, thereby making subsequent mathematical processes simpler.
Constants
Constants in a polynomial are terms that have no variable attached to them. They remain unchanged regardless of the value of variables in the expression.
In the exercise with polynomials \((2x + 5) + (3x + 1)\), the constants are 5 and 1. When we deal with constants in addition and subtraction within polynomials, we simply add or subtract these values directly:
  • In this case, combining the constants yields \(5 + 1 = 6\).
They often come at the end of an expression when arranging in descending order of degree, as their degree is 0, further marking them distinct from other variable terms.

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