Westonci.ca is your trusted source for finding answers to a wide range of questions, backed by a knowledgeable community. Get quick and reliable answers to your questions from a dedicated community of professionals on our platform. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
Certainly! Let's go through the solution step-by-step:
We start with the polynomial:
[tex]\[ -2 m^2 n^3 + 2 m^2 n^3 + 7 n^2 - 6 m^4 \][/tex]
Step 1: Simplify the Polynomial by Combining Like Terms
First, identify the like terms in the polynomial. We see that [tex]\(-2 m^2 n^3\)[/tex] and [tex]\(2 m^2 n^3\)[/tex] are like terms because they have the same powers of [tex]\(m\)[/tex] and [tex]\(n\)[/tex].
Combining these terms, we get:
[tex]\[ -2 m^2 n^3 + 2 m^2 n^3 = 0 \][/tex]
So, these two terms cancel each other out.
Step 2: Identify the Remaining Terms
After canceling the like terms, we are left with:
[tex]\[ 7 n^2 - 6 m^4 \][/tex]
Step 3: Determine the Degree of the Polynomial
We need to check the degree of each term:
- The term [tex]\(7 n^2\)[/tex] has a degree of 2 because the exponent of [tex]\(n\)[/tex] is 2.
- The term [tex]\(-6 m^4\)[/tex] has a degree of 4 because the exponent of [tex]\(m\)[/tex] is 4.
Since [tex]\(-6 m^4\)[/tex] is of degree 4, and [tex]\(7 n^2\)[/tex] is of lower degree, the overall degree of the polynomial, defined by the highest power, is 4.
Step 4: Confirm the Polynomial is a Binomial of Degree 4
A binomial consists of exactly two distinct terms. The simplified polynomial, [tex]\(7 n^2 - 6 m^4\)[/tex], has two distinct terms:
[tex]\[ 7 n^2 \quad \text{and} \quad -6 m^4 \][/tex]
Thus, it is a binomial.
For the polynomial to be a binomial of degree 4, we need to check the missing exponent on the [tex]\(m\)[/tex] in the initial term [tex]\(2 m^x n^3\)[/tex] to ensure that it forms a matching term with degree observations and combines correctly. The exponent [tex]\(x\)[/tex] on [tex]\(m\)[/tex] in [tex]\(m^x n^3\)[/tex] should make it equal degree term contributing correctly for the simplifying steps:
[tex]\[ -2 m^2 n^3 + 2 m^2 n^3 \][/tex]
Thus, x must have been 2 (since simplifying these terms correctly, cancel each other out if combining correctly into 0).
Thus, the answer is:
[tex]\[ \boxed{2} \][/tex]
We start with the polynomial:
[tex]\[ -2 m^2 n^3 + 2 m^2 n^3 + 7 n^2 - 6 m^4 \][/tex]
Step 1: Simplify the Polynomial by Combining Like Terms
First, identify the like terms in the polynomial. We see that [tex]\(-2 m^2 n^3\)[/tex] and [tex]\(2 m^2 n^3\)[/tex] are like terms because they have the same powers of [tex]\(m\)[/tex] and [tex]\(n\)[/tex].
Combining these terms, we get:
[tex]\[ -2 m^2 n^3 + 2 m^2 n^3 = 0 \][/tex]
So, these two terms cancel each other out.
Step 2: Identify the Remaining Terms
After canceling the like terms, we are left with:
[tex]\[ 7 n^2 - 6 m^4 \][/tex]
Step 3: Determine the Degree of the Polynomial
We need to check the degree of each term:
- The term [tex]\(7 n^2\)[/tex] has a degree of 2 because the exponent of [tex]\(n\)[/tex] is 2.
- The term [tex]\(-6 m^4\)[/tex] has a degree of 4 because the exponent of [tex]\(m\)[/tex] is 4.
Since [tex]\(-6 m^4\)[/tex] is of degree 4, and [tex]\(7 n^2\)[/tex] is of lower degree, the overall degree of the polynomial, defined by the highest power, is 4.
Step 4: Confirm the Polynomial is a Binomial of Degree 4
A binomial consists of exactly two distinct terms. The simplified polynomial, [tex]\(7 n^2 - 6 m^4\)[/tex], has two distinct terms:
[tex]\[ 7 n^2 \quad \text{and} \quad -6 m^4 \][/tex]
Thus, it is a binomial.
For the polynomial to be a binomial of degree 4, we need to check the missing exponent on the [tex]\(m\)[/tex] in the initial term [tex]\(2 m^x n^3\)[/tex] to ensure that it forms a matching term with degree observations and combines correctly. The exponent [tex]\(x\)[/tex] on [tex]\(m\)[/tex] in [tex]\(m^x n^3\)[/tex] should make it equal degree term contributing correctly for the simplifying steps:
[tex]\[ -2 m^2 n^3 + 2 m^2 n^3 \][/tex]
Thus, x must have been 2 (since simplifying these terms correctly, cancel each other out if combining correctly into 0).
Thus, the answer is:
[tex]\[ \boxed{2} \][/tex]
Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Westonci.ca is committed to providing accurate answers. Come back soon for more trustworthy information.
