Problem 2 Add the polynomials. $$\left(-... [FREE SOLUTION]

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Prealgebra

Julie Miller, Molly O鈥橬eill, Nancy Hyde

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Chapter 10: Problem 2

Add the polynomials. $$\left(-3 y^{5}-4 y^{2}+6 y\right)+\left(2 y^{5}+6 y^{2}-9 y\right)$$

Short Answer

Expert verified

-y^{5} + 2 y^{2} - 3 y

Step by step solution

- Identify Like Terms

In the given expression $(-3 y^{5}-4 y^{2}+6 y) + (2 y^{5}+6 y^{2}-9 y)$, identify the like terms. Like terms are the terms that have the same variable raised to the same power.

- Group Like Terms Together

Group the like terms together: \[-3 y^{5} + 2 y^{5}\], \[-4 y^{2} + 6 y^{2}\], and \[6 y - 9 y\].

- Add the Like Terms

Add the coefficients of the like terms together: \[-3 y^{5} + 2 y^{5} = (-3+2) y^{5} = -y^{5}\], \[-4 y^{2} + 6 y^{2} = (-4+6) y^{2} = 2 y^{2}\], and \[6 y - 9 y = (6-9) y = -3 y\].

- Write the Resulting Polynomial

Combine the resulting terms to write the final polynomial: $-y^{5} + 2 y^{2} - 3 y$.

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Key Concepts

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

like terms

Like terms are terms in an algebraic expression that have the same variable raised to the same power.
For example, in the expression $-3y^{5}$ and \2y^{5}\ are like terms because they both contain \y\ to the power of 5.
Similarly, $-4y^{2}$ and \6y^{2}\ are like terms due to having \y\ squared, and \6y\ and $-9y$ are like terms because they both contain \y\ raised to the power of 1.
When solving polynomial addition, it's crucial to correctly identify like terms to combine them accurately.

coefficients

Coefficients are the numerical parts of terms that are multiplied by the variables.
To understand the difference, look at the terms from the original exercise:
- In $-3y^{5}$, -3 is the coefficient while \y^{5}\ is the variable part.
- In \2y^{5}\, 2 is the coefficient with the same variable part, \y^{5}\.
Understanding coefficients helps in adding like terms easily.
When we add the terms $-3y^{5} + 2y^{5}$, we are essentially adding the coefficients $-3 + 2$, resulting in \ -y^{5} \.

grouping terms

Grouping terms in polynomial addition means organizing the expression so that all like terms are together.
This makes the process of adding coefficients straightforward.
In the given problem, first identify all like terms: $-3y^{5}+2y^{5}$, $-4y^{2}+6y^{2}$, and \6y-9y\.
By grouping these like terms together, you isolate the coefficients and variables, making the addition easier to handle.
This step is essential for simplifying polynomial expressions accurately and efficiently.
After grouping terms, you simply add the coefficients of the like terms, and combine the resulting expressions to form the final polynomial.

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91影视

Add the polynomials. $$\left(-3 y^{5}-4 y^{2}+6 y\right)+\left(2 y^{5}+6 y^{2}-9 y\right)$$

Short Answer

Step by step solution

- Identify Like Terms

- Group Like Terms Together

- Add the Like Terms

- Write the Resulting Polynomial

Key Concepts

like terms

coefficients

grouping terms

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