Which statement is true about the polynomial
–10m4n3 + 8m2n6 + 3m4n3 – 2m2n6 – 6m2n6 after it has been fully simplified?
A It is a monomial with a degree of 4.
B It is a monomial with a degree of 7.
C It is a binomial with a degree of 6.
D It is a binomial with a degree of 8.
The Correct Answer and Explanation is :
The correct answer is: B It is a monomial with a degree of 7
To simplify the polynomial (-10m^4n^3 + 8m^2n^6 + 3m^4n^3 – 2m^2n^6 – 6m^2n^6), we first need to combine like terms. Like terms are those that have the same variables raised to the same powers.
- Identify Like Terms:
- The terms (-10m^4n^3) and (3m^4n^3) are like terms because they both contain (m^4n^3).
- The terms (8m^2n^6), (-2m^2n^6), and (-6m^2n^6) are also like terms because they all contain (m^2n^6).
- Combine the Like Terms:
- For (m^4n^3):
[
-10m^4n^3 + 3m^4n^3 = (-10 + 3)m^4n^3 = -7m^4n^3
] - For (m^2n^6):
[
8m^2n^6 – 2m^2n^6 – 6m^2n^6 = (8 – 2 – 6)m^2n^6 = 0m^2n^6
]
Since the combined result is (0m^2n^6), these terms cancel out entirely.
- Write the Fully Simplified Polynomial:
After simplifying, the polynomial becomes:
[
-7m^4n^3
] - Determine the Characteristics of the Simplified Polynomial:
The simplified polynomial (-7m^4n^3) is a monomial because it consists of a single term. To find its degree, we add the exponents of the variables. Here, the degree is calculated as follows:
- The exponent of (m) is (4).
- The exponent of (n) is (3).
[
\text{Degree} = 4 + 3 = 7
]
- Conclusion:
Based on the simplification, we have a monomial with a degree of (7). Therefore, the correct answer is B: It is a monomial with a degree of 7.
In summary, after fully simplifying the given polynomial, it reduces to a single monomial term, demonstrating the importance of recognizing and combining like terms in polynomial expressions.