Question

which polynomials are in standard form?

(x^2y^3 + y + 3xy^2)
(-5a^3 + 12a^2b - 15ab^2 + b^3)
(4xy + 2x^2y^2 + xy^3)
(7x^4 + 4x^3y - 3x^2y^2 - y^4)
(14b^3 + ab^3 - 6ab + 8ab^2)
(3a^4 + 4a^3b - 6a^2b^2 - 4ab^3 - b^4)

Explanation:

🆕 New Concept Discovered: Polynomials in Standard Form
Arranging terms by degree in alphabetical order

Step 1: Understand standard form for multivariable polynomials

For polynomials with more than one variable, standard form is achieved by:

1. Choosing a principal variable (usually the one that comes first alphabetically, like \(a\) in terms of \(a\) and \(b\), or \(x\) in terms of \(x\) and \(y\)).
2. Ordering the terms in descending order of the powers of that principal variable.
3. If powers of the principal variable are equal, we look at the powers of the next variable in alphabetical order.

Let's analyze each option based on these rules.

Step 2: Analyze each option

• Option 1: \(x^2y^3 + y + 3xy^2\)
• Let's look at the powers of \(x\):
• First term: \(x^2y^3\) (power of \(x\) is 2)
• Second term: \(y\) (power of \(x\) is 0)
• Third term: \(3xy^2\) (power of \(x\) is 1)
• The powers of \(x\) go from \(2

ightarrow 0
ightarrow 1\). This is not in descending order.

• Not in standard form.

• Option 2: \(-5a^3 + 12a^2b - 15ab^2 + b^3\)
• Let's look at the powers of \(a\):
• First term: \(-5a^3\) (power of \(a\) is 3)
• Second term: \(12a^2b\) (power of \(a\) is 2)
• Third term: \(-15ab^2\) (power of \(a\) is 1)
• Fourth term: \(b^3\) (power of \(a\) is 0)
• The powers of \(a\) decrease perfectly: \(3

ightarrow 2
ightarrow 1
ightarrow 0\).

• In standard form.

• Option 3: \(4xy + 2x^2y^2 + xy^3\)
• Let's look at the powers of \(x\):
• First term: \(4xy\) (power of \(x\) is 1)
• Second term: \(2x^2y^2\) (power of \(x\) is 2)
• Third term: \(xy^3\) (power of \(x\) is 1)
• The powers of \(x\) go from \(1

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

• Not in standard form.

• Option 4: \(7x^4 + 4x^3y - 3x^2y^2 - y^4\)
• Let's look at the powers of \(x\):
• First term: \(7x^4\) (power of \(x\) is 4)
• Second term: \(4x^3y\) (power of \(x\) is 3)
• Third term: \(-3x^2y^2\) (power of \(x\) is 2)
• Fourth term: \(-y^4\) (power of \(x\) is 0)
• The powers of \(x\) decrease perfectly: \(4

ightarrow 3
ightarrow 2
ightarrow 0\).

• In standard form.

• Option 5: \(14b^3 + ab^3 - 6ab + 8ab^2\)
• Let's look at the powers of \(a\):
• First term: \(14b^3\) (power of \(a\) is 0)
• Second term: \(ab^3\) (power of \(a\) is 1)
• The powers of \(a\) are not in descending order.
• Not in standard form.

• Option 6: \(3a^4 + 4a^3b - 6a^2b^2 - 4ab^3 - b^4\)
• Let's look at the powers of \(a\):
• First term: \(3a^4\) (power of \(a\) is 4)
• Second term: \(4a^3b\) (power of \(a\) is 3)
• Third term: \(-6a^2b^2\) (power of \(a\) is 2)
• Fourth term: \(-4ab^3\) (power of \(a\) is 1)
• Fifth term: \(-b^4\) (power of \(a\) is 0)
• The powers of \(a\) decrease perfectly: \(4

ightarrow 3
ightarrow 2
ightarrow 1
ightarrow 0\).

• In standard form.

Answer:

The polynomials in standard form are:

• \(-5a^3 + 12a^2b - 15ab^2 + b^3\) (Option 2)
• \(7x^4 + 4x^3y - 3x^2y^2 - y^4\) (Option 4)
• \(3a^4 + 4a^3b - 6a^2b^2 - 4ab^3 - b^4\) (Option 6)