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Chapter 3: Problem 3

Add and write the resulting polynomial in descending order of degree. $$(7 y-4)+(2 y-1)$$

Short Answer

Expert verified
Answer: The resulting polynomial is \(9y - 5\).

Step by step solution

01

Identify Like Terms

In the given polynomials, the like terms are \(7y\) and \(2y\), and \(-4\) and \(-1\). We will add these pairs of like terms together.
02

Combine Like Terms

To combine the like terms, we simply add the coefficients of each term. So, we have \((7y + 2y) = (7+2)y = 9y\), and \((-4 + (-1)) = -4-1 = -5.\)
03

Write the Resulting Polynomial

Now that we've combined the like terms, we can write the resulting polynomial, which is \(9y - 5\).
04

Ensure Order is in Descending Degree

The resulting polynomial is already in descending order of degree, as \(y^1\) is the highest degree term. So, the final polynomial is \(\boxed{9y - 5}\).

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Key Concepts

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

Like Terms
When working with polynomials, it's important to understand the concept of like terms. Like terms have the same variable raised to the same power. In the expression
  • \(7y\) and \(2y\) are like terms because they both have the variable \(y\) raised to the power of 1.
  • Similarly, the constants \(-4\) and \(-1\) are like terms because they don鈥檛 contain a variable and are simply numbers.
To simplify polynomials, add or subtract only the like terms. This makes calculations easier and helps to maintain an organized expression.
Coefficients
The coefficient is the numerical factor in a term of a polynomial. It multiplies the variable it is attached to. In our exercise,
  • For \(7y\), the coefficient is \(7\).
  • For \(2y\), the coefficient is \(2\).
  • In constant terms like \(-4\) and \(-1\), there are no variables, so they can be considered coefficients themselves.
To combine terms, add or subtract coefficients of like terms. For example, adding \(7y\) and \(2y\) gives you \((7+2)y = 9y\). This is because the coefficients \(7\) and \(2\) are added together.
Descending Order
Arranging terms in descending order in a polynomial means ordering them by the degree from highest to lowest. The degree of a term is defined by the exponent of its variable.
  • For instance, in \(9y - 5\), the term \(9y\) has a degree of 1, which is higher than the degree of \(-5\) (which is degree 0).
When writing polynomials, place terms with higher powers first to follow mathematical convention. This makes the polynomial more readable and standardizes its presentation.
Terms of a Polynomial
A polynomial is made up of terms, where each term can be a constant or a variable raised to a power, multiplied by a coefficient. In the example given,
  • \(7y\) and \(2y\) are each separate terms.
  • The constant term \(-4\) is not linked to any variable.
Each term is separated by an addition or subtraction sign. When combining polynomials, you focus on these individual terms and rearrange or simplify them as shown. Writing in clear terms is essential for keeping your work organized.

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Most popular questions from this chapter

List all factors of each number. $$40$$

Complete the table. The first row will contain variable expressions. only. To find the sum, add the two given expressions. To find the difference, subtract the second expression from the first. After finding the variable expressions for the sum and difference, evaluate each variable expression using the given values. $$\begin{array}{|r|l|l|l|l} & \text { Expression 1 } & \text { Expression 2 } & \text { Sum } & \text { Difference } \\ \hline y & 2 y^{2}-5 y-3 & 3 y^{2}-5 y+2 & & \\ \hline 5 & & & & \\ \hline-1 & & & & \\ \hline \end{array}$$

Find the prime factorization. Write the answer in exponential form. $$220$$

Identify the terms in the polynomial and their coefficients. $$6 t-1$$

Translate to a variable expression. twenty less than a number
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