Question

which polynomial is in standard form?

\\(x^4 + 3x^3y - 5xy^3 + y^4\\)

\\(-x^4 + x^3y^2 + 7xy^3 - 2y^4x^2\\)

\\(8a^3 + 10ab^2 - 12a^2b^3\\)

Explanation:

🆕 New Concept Discovered: Standard Form of Multivariable Polynomials
Arranging terms by descending powers of one variable

Step 1: Understand standard form for multivariable polynomials

For a polynomial with more than one variable (like \(x\) and \(y\)), the standard form is typically written by choosing one variable (usually alphabetical, like \(x\)) and arranging the terms so that the exponents of that variable decrease from left to right.

If we choose \(x\) as our primary variable:

• The powers of \(x\) should go down: \(x^4

ightarrow x^3
ightarrow x^2
ightarrow x^1
ightarrow \text{no } x\).

• Within each term, variables are written in alphabetical order (e.g., \(x^2y^4\) instead of \(y^4x^2\)).

Step 2: Analyze the first option

Let's look at the first polynomial:

$$ x^4 + 3x^3y - 5xy^3 + y^4 $$

• The powers of \(x\) in each term from left to right are:
• First term: \(x^4\) (power of \(4\))
• Second term: \(3x^3y\) (power of \(3\))
• Third term: \(-5xy^3\) (power of \(1\))
• Fourth term: \(y^4\) (power of \(0\))
• The powers of \(x\) decrease consistently: \(4

ightarrow 3
ightarrow 1
ightarrow 0\).

• The variables in each term are written in alphabetical order (\(x\) before \(y\)).

This polynomial is in standard form.

Step 3: Analyze the other options to verify

• Second option: \(-x^4 + x^3y^2 + 7xy^3 - 2y^4x^2\)
• The last term has the variables out of alphabetical order (\(y^4x^2\) instead of \(x^2y^4\)).
• If we rewrite it alphabetically as \(-2x^2y^4\), the powers of \(x\) would go: \(4

ightarrow 3
ightarrow 1
ightarrow 2\), which is not in descending order.

• Third option: \(8a^3 + 10ab^2 - 12a^2b^3\)
• The powers of \(a\) are: \(3

ightarrow 1
ightarrow 2\). This is not in descending order.

Answer:

\( x^4 + 3x^3y - 5xy^3 + y^4 \)