poopey
Answered

Get the answers you need at Westonci.ca, where our expert community is dedicated to providing you with accurate information. Experience the ease of finding reliable answers to your questions from a vast community of knowledgeable experts. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.

Which polynomial is in standard form?

A. [tex]3xy + 6x^3y^2 - 4x^4y^3 + 19x^7y^4[/tex]

B. [tex]18x^5 - 7x^2y - 2xy^2 + 17y^4[/tex]

C. [tex]x^5y^5 - 3xy - 11x^2y^2 + 12[/tex]

D. [tex]15 + 12xy^2 - 11x^9y^5 + 5x^7y^2[/tex]

Sagot :

GinnyAnswer
To determine which polynomial is in standard form, let's break down the process and analyze each polynomial step by step. A polynomial in standard form is one where the terms are arranged in descending order of their total degrees.

The total degree of a term is the sum of the exponents of all the variables in that term.

### Analyzing Polynomial 1:
[tex]\[ P_1 = 3xy + 6x^3 y^2 - 4x^4 y^3 + 19x^7 y^4 \][/tex]

- \(3xy\): Total degree = \(1 + 1 = 2\)
- \(6x^3 y^2\): Total degree = \(3 + 2 = 5\)
- \(4x^4 y^3\): Total degree = \(4 + 3 = 7\)
- \(19x^7 y^4\): Total degree = \(7 + 4 = 11\)

Degrees of terms: \([2, 5, 7, 11]\)

### Analyzing Polynomial 2:
[tex]\[ P_2 = 18x^5 - 7x^2 y - 2xy^2 + 17y^4 \][/tex]

- \(18x^5\): Total degree = \(5\)
- \(-7x^2 y\): Total degree = \(2 + 1 = 3\)
- \(-2xy^2\): Total degree = \(1 + 2 = 3\)
- \(17y^4\): Total degree = \(4\)

Degrees of terms: \([5, 3, 3, 4]\)

### Analyzing Polynomial 3:
[tex]\[ P_3 = x^5 y^5 - 3xy - 11x^2 y^2 + 12 \][/tex]

- \(x^5 y^5\): Total degree = \(5 + 5 = 10\)
- \(-3xy\): Total degree = \(1 + 1 = 2\)
- \(-11x^2 y^2\): Total degree = \(2 + 2 = 4\)
- \(12\): Total degree = \(0\)

Degrees of terms: \([10, 2, 4, 0]\)

### Analyzing Polynomial 4:
[tex]\[ P_4 = 15 + 12xy^2 - 11x^9 y^5 + 5x^7 y^2 \][/tex]

- \(15\): Total degree = \(0\)
- \(12xy^2\): Total degree = \(1 + 2 = 3\)
- \(-11x^9 y^5\): Total degree = \(9 + 5 = 14\)
- \(5x^7 y^2\): Total degree = \(7 + 2 = 9\)

Degrees of terms: \([0, 3, 14, 9]\)

### Conclusion:
By checking if the degrees are in descending order for each polynomial:

- For \( P_1 \): The degrees \([2, 5, 7, 11]\) are in ascending order, not descending.
- For \( P_2 \): The degrees \([5, 3, 3, 4]\) are not in descending order.
- For \( P_3 \): The degrees \([10, 2, 4, 0]\) are not in descending order.
- For \( P_4 \): The degrees \([0, 3, 14, 9]\) are not in descending order.

None of the given polynomials are in standard form. Therefore, the correct answer is that no polynomial is in standard form.

Thus, the result is:
[tex]\[ \boxed{\text{None}} \][/tex]
We appreciate your time. Please revisit us for more reliable answers to any questions you may have. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Westonci.ca is committed to providing accurate answers. Come back soon for more trustworthy information.