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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 :

GinnyAnsweridn
To determine which polynomial is in standard form, we need to identify the polynomial that is organized in descending order of the powers of the variables. Let's analyze each polynomial step-by-step.

1. Polynomial 1: [tex]\(3 x y + 6 x^3 y^2 - 4 x^4 y^3 + 19 x^7 y^4\)[/tex]

For each term, we can calculate the total power of [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
- [tex]\(3 x y\)[/tex] has [tex]\(1+1 = 2\)[/tex]
- [tex]\(6 x^3 y^2\)[/tex] has [tex]\(3+2 = 5\)[/tex]
- [tex]\(-4 x^4 y^3\)[/tex] has [tex]\(4+3 = 7\)[/tex]
- [tex]\(19 x^7 y^4\)[/tex] has [tex]\(7+4 = 11\)[/tex]

The terms written in descending order of total power are:
[tex]\[ 19 x^7 y^4, -4 x^4 y^3, 6 x^3 y^2, 3 x y \][/tex]
However, it does not exactly adhere to standard form when considering individual variable powers.

2. Polynomial 2: [tex]\(18 x^5 - 7 x^2 y - 2 x y^2 + 17 y^4\)[/tex]

The order for each term descending:
- [tex]\(18 x^5\)[/tex]
- [tex]\(-7 x^2 y\)[/tex]
- [tex]\(-2 x y^2\)[/tex]
- [tex]\(17 y^4\)[/tex]

These terms appear to be ordered correctly in terms of descending power combinations:
[tex]\[18 x^5, -7 x^2 y, -2 x y^2, 17 y^4\][/tex]

3. Polynomial 3: [tex]\(x^5 y^5 - 3 x y - 11 x^2 y^2 + 12\)[/tex]

For each term in descending order:
- [tex]\(x^5 y^5\)[/tex] has [tex]\(5+5 = 10\)[/tex]
- [tex]\(-3 x y\)[/tex] has [tex]\(1+1 = 2\)[/tex]
- [tex]\(-11 x^2 y^2\)[/tex] has [tex]\(2+2 = 4\)[/tex]
- [tex]\(12\)[/tex] has the constant power 0

Therefore, we have:
[tex]\[x^5 y^5, -11 x^2 y^2, -3 x y, 12\][/tex]
The combination powers are inconsistent when adding the constant part.

4. Polynomial 4: [tex]\(15 + 12 x y^2 - 11 x^9 y^5 + 5 x^7 y^2\)[/tex]

In descending order:
- [tex]\(15\)[/tex] has the constant 0
- [tex]\(12 x y^2\)[/tex] has [tex]\(1+2 = 3\)[/tex]
- [tex]\(5 x^7 y^2\)[/tex] has [tex]\(7+2 = 9\)[/tex]
- [tex]\(-11 x^9 y^5\)[/tex] has [tex]\(9+5 = 14\)[/tex]

If we order these by total variable powers, they would return:
[tex]\[-11 x^9 y^5, 5 x^7 y^2, 12 x y^2, 15 \][/tex]

After reviewing, the polynomial that properly orders the terms by descending powers of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] is:

[tex]\[ 18 x^5 - 7 x^2 y - 2 x y^2 + 17 y^4 \][/tex]

Thus, the polynomial in standard form is:

[tex]\[ \boxed{2} \][/tex]
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