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Julian fully simplifies this polynomial and then writes it in standard form.

[tex]\[ 4x^2 y^2 - 2y^4 - 8xy^3 + 9x^3 y + 6y^4 - 2xy^3 - 3x^4 + x^2 y^2 \][/tex]

If Julian wrote the last term as [tex]\(-3x^4\)[/tex], which must be the first term of his polynomial in standard form?

A. [tex]\(4y^4\)[/tex]
B. [tex]\(6y^4\)[/tex]
C. [tex]\(-2xy^3\)[/tex]
D. [tex]\(-10xy^3\)[/tex]

Sagot :

GinnyAnswer
Let's start by fully simplifying the given polynomial by combining like terms:

The polynomial is:
[tex]\[ 4x^2y^2 - 2y^4 - 8xy^3 + 9x^3y + 6y^4 - 2xy^3 - 3x^4 + x^2y^2 \][/tex]

First, we group like terms together:

### Grouping the like terms:

1. [tex]\(x^4\)[/tex] terms:
[tex]\[ -3x^4 \][/tex]

2. [tex]\(x^3y\)[/tex] terms:
[tex]\[ 9x^3y \][/tex]

3. [tex]\(x^2y^2\)[/tex] terms:
[tex]\[ 4x^2y^2 + x^2y^2 = 5x^2y^2 \][/tex]

4. [tex]\(xy^3\)[/tex] terms:
[tex]\[ -8xy^3 - 2xy^3 = -10xy^3 \][/tex]

5. [tex]\(y^4\)[/tex] terms:
[tex]\[ -2y^4 + 6y^4 = 4y^4 \][/tex]

So, the simplified polynomial is:
[tex]\[ -3x^4 + 9x^3y + 5x^2y^2 - 10xy^3 + 4y^4 \][/tex]

Julian wrote the last term as [tex]\( -3x^4 \)[/tex]. In standard form, the polynomial is arranged in descending order of the powers of x and y. Therefore, the first term should be the one with the highest power of y, which is:

[tex]\[ 4y^4 \][/tex]

Thus, the correct answer is:

[tex]\[ 4y^4 \][/tex]
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