Answer :
To determine which polynomials are in standard form, we need to ensure that for each polynomial, its monomials are arranged in descending order of their degrees, with similar variables grouped together. Let's analyze each given polynomial step by step.
1. \(x^2 y^3 + y + 3 x y^2\)
- Monomials: \(x^2 y^3\), \(y\), \(3 x y^2\)
- Degrees: \(5\), \(1\), \(3\)
- Arranged: \(x^2 y^3\) (degree 5), \(3 x y^2\) (degree 3), \(y\) (degree 1)
- This polynomial is not in standard form because the degrees are not in descending order.
2. \(-5 a^3 + 12 a^2 b - 15 a b^2 + b^3\)
- Monomials: \(-5 a^3\), \(12 a^2 b\), \(-15 a b^2\), \(b^3\)
- Degrees: \(3\), \(3\), \(3\), \(3\)
- Arranged: \(-5 a^3\) (degree 3), \(12 a^2 b\) (degree 3), \(-15 a b^2\) (degree 3), \(b^3\) (degree 3)
- This polynomial is in standard form because all degrees are essentially the same and the sequence does follow a pattern.
3. \(4 x y + 2 x^2 y^2 + x y^3\)
- Monomials: \(4 x y\), \(2 x^2 y^2\), \(x y^3\)
- Degrees: \(2\), \(4\), \(4\)
- Arranged: \(2 x^2 y^2\) (degree 4), \(x y^3\) (degree 4), \(4 x y\) (degree 2)
- This polynomial is not in standard form because the degrees are not in descending order.
4. \(7 x^4 + 4 x^3 y - 3 x^2 y^2 - y^4\)
- Monomials: \(7 x^4\), \(4 x^3 y\), \(-3 x^2 y^2\), \(-y^4\)
- Degrees: \(4\), \(4\), \(4\), \(4\)
- Arranged: \(7 x^4\) (degree 4), \(4 x^3 y\) (degree 4), \(-3 x^2 y^2\) (degree 4), \(-y^4\) (degree 4)
- This polynomial is in standard form because all degrees are essentially the same and the sequence follows a pattern of descending order.
5. \(14 b^3 + a b^3 - 6 a b + 8 a b^2\)
- Monomials: \(14 b^3\), \(a b^3\), \(-6 a b\), \(8 a b^2\)
- Degrees: \(3\), \(4\), \(2\), \(3\)
- Arranged: \( a b^3\) (degree 4), \(14 b^3\) (degree 3), \(8 a b^2\) (degree 3), \(-6 a b\) (degree 2)
- This polynomial is in standard form because all degrees arranged in descending order: \(4\), \(3\), \(3\), then \(2\).
6. \(3 a^4 + 4 a^3 b - 6 a^2 b^2 - 4 a b^3 - b^4\)
- Monomials: \(3 a^4\), \(4 a^3 b\), \(-6 a^2 b^2\), \(-4 a b^3\), \(-b^4\)
- Degrees: \(4\), \(4\), \(4\), \(4\), \(4\)
- Arranged: \(3 a^4\) (degree 4), \(4 a^3 b\) (degree 4), \(-6 a^2 b^2\) (degree 4), \(-4 a b^3\) (degree 4), \(-b^4\) (degree 4)
- This polynomial is in standard form as follows a pattern of descending order, even if only one degree.
Thus, the polynomials in standard form are:
- \(-5 a^3 + 12 a^2 b - 15 a b^2 + b^3\)
- \(7 x^4 + 4 x^3 y - 3 x^2 y^2 - y^4\)
- \(14 b^3 + a b^3 - 6 a b + 8 a b^2\)
- \(3 a^4 + 4 a^3 b - 6 a^2 b^2 - 4 a b^3 - b^4\)
So, the polynomials in standard form are the second, fourth, fifth, and sixth polynomials.
1. \(x^2 y^3 + y + 3 x y^2\)
- Monomials: \(x^2 y^3\), \(y\), \(3 x y^2\)
- Degrees: \(5\), \(1\), \(3\)
- Arranged: \(x^2 y^3\) (degree 5), \(3 x y^2\) (degree 3), \(y\) (degree 1)
- This polynomial is not in standard form because the degrees are not in descending order.
2. \(-5 a^3 + 12 a^2 b - 15 a b^2 + b^3\)
- Monomials: \(-5 a^3\), \(12 a^2 b\), \(-15 a b^2\), \(b^3\)
- Degrees: \(3\), \(3\), \(3\), \(3\)
- Arranged: \(-5 a^3\) (degree 3), \(12 a^2 b\) (degree 3), \(-15 a b^2\) (degree 3), \(b^3\) (degree 3)
- This polynomial is in standard form because all degrees are essentially the same and the sequence does follow a pattern.
3. \(4 x y + 2 x^2 y^2 + x y^3\)
- Monomials: \(4 x y\), \(2 x^2 y^2\), \(x y^3\)
- Degrees: \(2\), \(4\), \(4\)
- Arranged: \(2 x^2 y^2\) (degree 4), \(x y^3\) (degree 4), \(4 x y\) (degree 2)
- This polynomial is not in standard form because the degrees are not in descending order.
4. \(7 x^4 + 4 x^3 y - 3 x^2 y^2 - y^4\)
- Monomials: \(7 x^4\), \(4 x^3 y\), \(-3 x^2 y^2\), \(-y^4\)
- Degrees: \(4\), \(4\), \(4\), \(4\)
- Arranged: \(7 x^4\) (degree 4), \(4 x^3 y\) (degree 4), \(-3 x^2 y^2\) (degree 4), \(-y^4\) (degree 4)
- This polynomial is in standard form because all degrees are essentially the same and the sequence follows a pattern of descending order.
5. \(14 b^3 + a b^3 - 6 a b + 8 a b^2\)
- Monomials: \(14 b^3\), \(a b^3\), \(-6 a b\), \(8 a b^2\)
- Degrees: \(3\), \(4\), \(2\), \(3\)
- Arranged: \( a b^3\) (degree 4), \(14 b^3\) (degree 3), \(8 a b^2\) (degree 3), \(-6 a b\) (degree 2)
- This polynomial is in standard form because all degrees arranged in descending order: \(4\), \(3\), \(3\), then \(2\).
6. \(3 a^4 + 4 a^3 b - 6 a^2 b^2 - 4 a b^3 - b^4\)
- Monomials: \(3 a^4\), \(4 a^3 b\), \(-6 a^2 b^2\), \(-4 a b^3\), \(-b^4\)
- Degrees: \(4\), \(4\), \(4\), \(4\), \(4\)
- Arranged: \(3 a^4\) (degree 4), \(4 a^3 b\) (degree 4), \(-6 a^2 b^2\) (degree 4), \(-4 a b^3\) (degree 4), \(-b^4\) (degree 4)
- This polynomial is in standard form as follows a pattern of descending order, even if only one degree.
Thus, the polynomials in standard form are:
- \(-5 a^3 + 12 a^2 b - 15 a b^2 + b^3\)
- \(7 x^4 + 4 x^3 y - 3 x^2 y^2 - y^4\)
- \(14 b^3 + a b^3 - 6 a b + 8 a b^2\)
- \(3 a^4 + 4 a^3 b - 6 a^2 b^2 - 4 a b^3 - b^4\)
So, the polynomials in standard form are the second, fourth, fifth, and sixth polynomials.
