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Marcus states that the polynomial expression [tex][tex]$3x^3 - 4x^2y + y^3 + 2$[/tex][/tex] is in standard form. Ariel states that it should be [tex][tex]$y^3 - 4x^2y + 3x^3 + 2$[/tex][/tex]. Explain which student is correct and why.

Sagot :

GinnyAnswer
Certainly! Let's analyze the polynomial expressions provided by Marcus and Ariel to determine which one follows the standard form for polynomials.

### Understanding Polynomial Standard Form

For a polynomial in multiple variables, there are conventions for how to order the terms:

1. Order by Powers of One Variable First: Typically, terms are ordered primarily by the descending powers of one variable.
2. Resolve Ties by Secondary Variables: If the powers of the primary variable are the same, we then order by the descending powers of another variable.

### Analyzing Marcus's Expression

Marcus's polynomial is:
[tex]\[ 3x^3 - 4x^2y + y^3 + 2 \][/tex]

### Analyzing Ariel's Expression

Ariel's polynomial is:
[tex]\[ y^3 - 4x^2y + 3x^3 + 2 \][/tex]

### Compare Both Expressions by Conventional Ordering

1. Marcus's Ordering:
[tex]\[ 3x^3 - 4x^2y + y^3 + 2 \][/tex]
- Terms in Marcus's version are:
- [tex]\( 3x^3 \)[/tex]
- [tex]\( -4x^2y \)[/tex]
- [tex]\( y^3 \)[/tex]
- [tex]\( +2 \)[/tex]
- Ordered by descending powers of [tex]\( x \)[/tex] first:
- [tex]\( 3x^3 \)[/tex]
- [tex]\( -4x^2y \)[/tex]
- [tex]\( y^3 \)[/tex]
- [tex]\( +2 \)[/tex]

2. Ariel's Ordering:
[tex]\[ y^3 - 4x^2y + 3x^3 + 2 \][/tex]
- Terms in Ariel's version are:
- [tex]\( y^3 \)[/tex]
- [tex]\( -4x^2y \)[/tex]
- [tex]\( 3x^3 \)[/tex]
- [tex]\( +2 \)[/tex]
- Ordered by descending powers of [tex]\( y \)[/tex] first:
- [tex]\( y^3 \)[/tex]
- [tex]\( -4x^2y \)[/tex]
- [tex]\( 3x^3 \)[/tex]
- [tex]\( +2 \)[/tex]

### Conventional Standard Form

The conventional way to write polynomials, especially in contexts where one variable is typically given precedence, is to order by the descending powers of [tex]\( x \)[/tex] first.

### Conclusion

Given that the conventional ordering is generally by descending powers of [tex]\( x \)[/tex] first, Marcus's ordering of the polynomial:
[tex]\[ 3x^3 - 4x^2y + y^3 + 2 \][/tex]
is the correct way to write the polynomial in standard form.

Hence, Marcus is correct because he ordered the polynomial in descending powers of [tex]\( x \)[/tex], which aligns with the common convention for writing polynomials.
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