Which Polynomial Correctly Combines The Like Terms And Expresses The Given Polynomial In Standard Form?Given Polynomial: ${ 8mn^5 - 2m^6 + 5m 2n 4 - M 3n 3 + N^6 - 4m^6 + 9m 2n 4 - Mn^5 - 4m 3n 3 }$Options:A. $[ N^6 + 7mn^5 + 14m 2n 4 -

Understanding the Concept of Like Terms

In algebra, like terms are those terms that have the same variable(s) raised to the same power. When combining like terms, we add or subtract their coefficients. For example, in the expression $3x^2 + 2x^2$, the like terms are $3x^2$ and $2x^2$, and when we combine them, we get $(3+2)x^2 = 5x^2$.

The Given Polynomial

The given polynomial is:

$$8mn^5 - 2m^6 + 5m^2n^4 - m^3n^3 + n^6 - 4m^6 + 9m^2n^4 - mn^5 - 4m^3n^3$$

Combining Like Terms

To express the given polynomial in standard form, we need to combine the like terms. We can start by grouping the terms with the same variable(s) raised to the same power.

Grouping Terms with the Same Variable(s) Raised to the Same Power

Let's group the terms with the same variable(s) raised to the same power:

• Terms with $n^6$: $n^6 - 4m^6$
• Terms with $m^6$: $-2m^6 - 4m^6$
• Terms with $m^2n^4$: $5m^2n^4 + 9m^2n^4$
• Terms with $m^3n^3$: $-m^3n^3 - 4m^3n^3$
• Terms with $mn^5$: $8mn^5 - mn^5$
• Terms with no variables: None

Combining Like Terms

Now, let's combine the like terms:

• $n^6 - 4m^6$ remains the same
• $-2m^6 - 4m^6 = -6m^6$
• $5m^2n^4 + 9m^2n^4 = 14m^2n^4$
• $-m^3n^3 - 4m^3n^3 = -5m^3n^3$
• $8mn^5 - mn^5 = 7mn^5$

Expressing the Polynomial in Standard Form

Now that we have combined the like terms, we can express the polynomial in standard form:

$$n^6 - 6m^6 + 14m^2n^4 - 5m^3n^3 + 7mn^5$$

Comparing with the Options

Let's compare the polynomial we obtained with the options:

• Option A: $n^6 + 7mn^5 + 14m^2n^4 - 5m^3n^3 - 6m^6$
• Option B: $n^6 + 7mn^5 + 14m^2n^4 - 5m^3n^3 - 6m^6$
• Option C: $n^6 + 7mn^5 + 14m^2n^4 - 5m^3n^3 - 6m^6$
• Option D: $n^6 + 7mn^5 + 14m^2n^4 - 5m^3n^3 - 6m^6$

Conclusion

Based on our calculations, the correct polynomial that combines the like terms and expresses the given polynomial in standard form is:

$$n^6 - 6m^6 + 14m^2n^4 - 5m^3n^3 + 7mn^5$$

This polynomial matches with option A.

Final Answer

The final answer is option A.

Q: What are like terms in algebra?

A: Like terms are those terms that have the same variable(s) raised to the same power. For example, in the expression $3x^2 + 2x^2$, the like terms are $3x^2$ and $2x^2$.

Q: How do I identify like terms in a polynomial?

A: To identify like terms, look for terms with the same variable(s) raised to the same power. For example, in the polynomial $2x^2 + 3x^2 + 4y^2 + 5y^2$, the like terms are $2x^2$ and $3x^2$, and $4y^2$ and $5y^2$.

Q: How do I combine like terms in a polynomial?

A: To combine like terms, add or subtract their coefficients. For example, in the expression $3x^2 + 2x^2$, the like terms are $3x^2$ and $2x^2$, and when we combine them, we get $(3+2)x^2 = 5x^2$.

Q: What is the standard form of a polynomial?

A: The standard form of a polynomial is when the terms are arranged in descending order of their exponents. For example, the standard form of the polynomial $2x^2 + 3x^2 + 4y^2 + 5y^2$ is $4y^2 + 5y^2 + 2x^2 + 3x^2$.

Q: How do I express a polynomial in standard form?

A: To express a polynomial in standard form, combine the like terms and arrange the terms in descending order of their exponents.

Q: What is the importance of combining like terms in algebra?

A: Combining like terms is important in algebra because it helps to simplify expressions and make them easier to work with. It also helps to identify patterns and relationships between variables.

Q: Can you give an example of a polynomial that needs to be combined?

A: Here is an example of a polynomial that needs to be combined:

$$2x^2 + 3x^2 + 4y^2 + 5y^2 + 6z^2 + 7z^2$$

To combine the like terms, we add or subtract their coefficients:

$$2x^2 + 3x^2 = 5x^2$$

$$4y^2 + 5y^2 = 9y^2$$

$$6z^2 + 7z^2 = 13z^2$$

So, the combined polynomial is:

$$5x^2 + 9y^2 + 13z^2$$

Q: Can you give an example of a polynomial that needs to be expressed in standard form?

A: Here is an example of a polynomial that needs to be expressed in standard form:

$$2x^2 + 3x^2 + 4y^2 + 5y^2 + 6z^2 + 7z^2 - 2x^2 - 3x^2 - 4y^2 - 5y^2 - 6z^2 - 7z^2$$

To express the polynomial in standard form, we combine the like terms and arrange the terms in descending order of their exponents:

$$2x^2 + 3x^2 - 2x^2 - 3x^2 = 0$$

$$4y^2 + 5y^2 - 4y^2 - 5y^2 = 0$$

$$6z^2 + 7z^2 - 6z^2 - 7z^2 = 0$$

So, the polynomial in standard form is:

$$0 + 0 + 0$$

However, this is not a very interesting polynomial! Let's try another example.

Q: Can you give another example of a polynomial that needs to be expressed in standard form?

A: Here is another example of a polynomial that needs to be expressed in standard form:

$$2x^2 + 3x^2 + 4y^2 + 5y^2 + 6z^2 + 7z^2 + 2x^2 + 3x^2 + 4y^2 + 5y^2 + 6z^2 + 7z^2$$

To express the polynomial in standard form, we combine the like terms and arrange the terms in descending order of their exponents:

$$2x^2 + 3x^2 + 2x^2 + 3x^2 = 8x^2$$

$$4y^2 + 5y^2 + 4y^2 + 5y^2 = 16y^2$$

$$6z^2 + 7z^2 + 6z^2 + 7z^2 = 26z^2$$

So, the polynomial in standard form is:

$$8x^2 + 16y^2 + 26z^2$$

Q: Can you give an example of a polynomial that needs to be combined and expressed in standard form?

A: Here is an example of a polynomial that needs to be combined and expressed in standard form:

$$2x^2 + 3x^2 + 4y^2 + 5y^2 + 6z^2 + 7z^2 - 2x^2 - 3x^2 - 4y^2 - 5y^2 - 6z^2 - 7z^2 + 2x^2 + 3x^2 + 4y^2 + 5y^2 + 6z^2 + 7z^2$$

To combine the like terms and express the polynomial in standard form, we follow the same steps as before:

$$2x^2 + 3x^2 - 2x^2 - 3x^2 + 2x^2 + 3x^2 = 8x^2$$

$$4y^2 + 5y^2 - 4y^2 - 5y^2 + 4y^2 + 5y^2 = 16y^2$$

$$6z^2 + 7z^2 - 6z^2 - 7z^2 + 6z^2 + 7z^2 = 26z^2$$

So, the polynomial in standard form is:

$$8x^2 + 16y^2 + 26z^2$$

Q: Can you give an example of a polynomial that needs to be combined and expressed in standard form, but with more complex terms?

A: Here is an example of a polynomial that needs to be combined and expressed in standard form, but with more complex terms:

$$2x^3 + 3x^3 + 4y^3 + 5y^3 + 6z^3 + 7z^3 - 2x^3 - 3x^3 - 4y^3 - 5y^3 - 6z^3 - 7z^3 + 2x^3 + 3x^3 + 4y^3 + 5y^3 + 6z^3 + 7z^3$$

To combine the like terms and express the polynomial in standard form, we follow the same steps as before:

$$2x^3 + 3x^3 - 2x^3 - 3x^3 + 2x^3 + 3x^3 = 8x^3$$

$$4y^3 + 5y^3 - 4y^3 - 5y^3 + 4y^3 + 5y^3 = 16y^3$$

$$6z^3 + 7z^3 - 6z^3 - 7z^3 + 6z^3 + 7z^3 = 26z^3$$

So, the polynomial in standard form is:

$$8x^3 + 16y^3 + 26z^3$$

Q: Can you give an example of a polynomial that needs to be combined and expressed in standard form, but with more complex terms and variables?

A: Here is an example of a polynomial that needs to be combined and expressed in standard form, but with more complex terms and variables:

$$2x^3y^2 + 3x^3y^2 + 4x^3y^2 + 5x^3y^2 + 6x^3y^2 + 7x^3y^2 - 2x^3y^2 - 3x^3y^2 - 4x^3y^2 - 5x^3y^2 - 6x^3y^2 - 7x^3y^2 + 2x^3y^2 + 3x^3y^2 + 4x^3y^2 + 5x^3y^2 + 6x^3y^2 + 7x^3y^2$$

To combine the like terms and express the polynomial in standard form, we follow the same steps as before:

$$2x^3y^2 + 3x^3y^2$$