Welcome to Westonci.ca, your go-to destination for finding answers to all your questions. Join our expert community today! Experience the ease of finding reliable answers to your questions from a vast community of knowledgeable experts. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
Sagot :
To determine which algebraic expression is a polynomial with a degree of 5, we need to identify the highest degree term in each expression. The degree of a term is the sum of the exponents of the variables in that term. The degree of the polynomial is the highest degree among all its terms.
Let's examine each expression step by step:
1. [tex]\(3x^5 + 8x^4 y^2 - 9x^3 y^3 - 6y^5\)[/tex]
- [tex]\(3x^5\)[/tex] has a degree of [tex]\(5\)[/tex] (since the exponent of [tex]\(x\)[/tex] is 5).
- [tex]\(8x^4 y^2\)[/tex] has a degree of [tex]\(6\)[/tex] (sum of exponents: [tex]\(4 + 2\)[/tex]).
- [tex]\(9x^3 y^3\)[/tex] has a degree of [tex]\(6\)[/tex] (sum of exponents: [tex]\(3 + 3\)[/tex]).
- [tex]\(-6y^5\)[/tex] has a degree of [tex]\(5\)[/tex] (since the exponent of [tex]\(y\)[/tex] is 5).
The highest degree term is [tex]\(8x^4 y^2\)[/tex] or [tex]\(9x^3 y^3\)[/tex], both with a degree of [tex]\(6\)[/tex]. Hence, the degree of this polynomial is [tex]\(6\)[/tex].
2. [tex]\(2x y^4 + 4x^2 y^3 - 6x^3 y^2 - 7x^4\)[/tex]
- [tex]\(2x y^4\)[/tex] has a degree of [tex]\(5\)[/tex] (sum of exponents: [tex]\(1 + 4\)[/tex]).
- [tex]\(4x^2 y^3\)[/tex] has a degree of [tex]\(5\)[/tex] (sum of exponents: [tex]\(2 + 3\)[/tex]).
- [tex]\(6x^3 y^2\)[/tex] has a degree of [tex]\(5\)[/tex] (sum of exponents: [tex]\(3 + 2\)[/tex]).
- [tex]\(-7x^4\)[/tex] has a degree of [tex]\(4\)[/tex] (since the exponent of [tex]\(x\)[/tex] is 4).
The highest degree term is [tex]\(2x y^4\)[/tex], [tex]\(4x^2 y^3\)[/tex], or [tex]\(6x^3 y^2\)[/tex], all with a degree of [tex]\(5\)[/tex]. Hence, the degree of this polynomial is [tex]\(5\)[/tex].
3. [tex]\(8y^6 + y^5 - 5x y^3 + 7x^2 y^2 - x^3 y - 6x^4\)[/tex]
- [tex]\(8y^6\)[/tex] has a degree of [tex]\(6\)[/tex] (since the exponent of [tex]\(y\)[/tex] is 6).
- [tex]\(y^5\)[/tex] has a degree of [tex]\(5\)[/tex] (since the exponent of [tex]\(y\)[/tex] is 5).
- [tex]\(-5x y^3\)[/tex] has a degree of [tex]\(4\)[/tex] (sum of exponents: [tex]\(1 + 3\)[/tex]).
- [tex]\(7x^2 y^2\)[/tex] has a degree of [tex]\(4\)[/tex] (sum of exponents: [tex]\(2 + 2\)[/tex]).
- [tex]\(-x^3 y\)[/tex] has a degree of [tex]\(4\)[/tex] (sum of exponents: [tex]\(3 + 1\)[/tex]).
- [tex]\(-6x^4\)[/tex] has a degree of [tex]\(4\)[/tex] (since the exponent of [tex]\(x\)[/tex] is 4).
The highest degree term is [tex]\(8y^6\)[/tex] with a degree of [tex]\(6\)[/tex]. Hence, the degree of this polynomial is [tex]\(6\)[/tex].
4. [tex]\(-6x y^5 + 5x^2 y^3 - x^3 y^2 + 2x^2 y^3 - 3x y^5\)[/tex]
- [tex]\(-6x y^5\)[/tex] has a degree of [tex]\(6\)[/tex] (sum of exponents: [tex]\(1 + 5\)[/tex]).
- [tex]\(5x^2 y^3\)[/tex] has a degree of [tex]\(5\)[/tex] (sum of exponents: [tex]\(2 + 3\)[/tex]).
- [tex]\(-x^3 y^2\)[/tex] has a degree of [tex]\(5\)[/tex] (sum of exponents: [tex]\(3 + 2\)[/tex]).
- [tex]\(2x^2 y^3\)[/tex] has a degree of [tex]\(5\)[/tex] (sum of exponents: [tex]\(2 + 3\)[/tex]).
- [tex]\(-3x y^5\)[/tex] has a degree of [tex]\(6\)[/tex] (sum of exponents: [tex]\(1 + 5\)[/tex]).
The highest degree term is [tex]\(-6x y^5\)[/tex] or [tex]\(-3x y^5\)[/tex], both with a degree of [tex]\(6\)[/tex]. Hence, the degree of this polynomial is [tex]\(6\)[/tex].
From our examination, the only polynomial with a degree of [tex]\(5\)[/tex] is:
[tex]\[ 2x y^4 + 4x^2 y^3 - 6x^3 y^2 - 7x^4 \][/tex]
Thus, the algebraic expression which is a polynomial with a degree of [tex]\(5\)[/tex] is:
[tex]\[ 2x y^4 + 4x^2 y^3 - 6x^3 y^2 - 7x^4 \][/tex]
Let's examine each expression step by step:
1. [tex]\(3x^5 + 8x^4 y^2 - 9x^3 y^3 - 6y^5\)[/tex]
- [tex]\(3x^5\)[/tex] has a degree of [tex]\(5\)[/tex] (since the exponent of [tex]\(x\)[/tex] is 5).
- [tex]\(8x^4 y^2\)[/tex] has a degree of [tex]\(6\)[/tex] (sum of exponents: [tex]\(4 + 2\)[/tex]).
- [tex]\(9x^3 y^3\)[/tex] has a degree of [tex]\(6\)[/tex] (sum of exponents: [tex]\(3 + 3\)[/tex]).
- [tex]\(-6y^5\)[/tex] has a degree of [tex]\(5\)[/tex] (since the exponent of [tex]\(y\)[/tex] is 5).
The highest degree term is [tex]\(8x^4 y^2\)[/tex] or [tex]\(9x^3 y^3\)[/tex], both with a degree of [tex]\(6\)[/tex]. Hence, the degree of this polynomial is [tex]\(6\)[/tex].
2. [tex]\(2x y^4 + 4x^2 y^3 - 6x^3 y^2 - 7x^4\)[/tex]
- [tex]\(2x y^4\)[/tex] has a degree of [tex]\(5\)[/tex] (sum of exponents: [tex]\(1 + 4\)[/tex]).
- [tex]\(4x^2 y^3\)[/tex] has a degree of [tex]\(5\)[/tex] (sum of exponents: [tex]\(2 + 3\)[/tex]).
- [tex]\(6x^3 y^2\)[/tex] has a degree of [tex]\(5\)[/tex] (sum of exponents: [tex]\(3 + 2\)[/tex]).
- [tex]\(-7x^4\)[/tex] has a degree of [tex]\(4\)[/tex] (since the exponent of [tex]\(x\)[/tex] is 4).
The highest degree term is [tex]\(2x y^4\)[/tex], [tex]\(4x^2 y^3\)[/tex], or [tex]\(6x^3 y^2\)[/tex], all with a degree of [tex]\(5\)[/tex]. Hence, the degree of this polynomial is [tex]\(5\)[/tex].
3. [tex]\(8y^6 + y^5 - 5x y^3 + 7x^2 y^2 - x^3 y - 6x^4\)[/tex]
- [tex]\(8y^6\)[/tex] has a degree of [tex]\(6\)[/tex] (since the exponent of [tex]\(y\)[/tex] is 6).
- [tex]\(y^5\)[/tex] has a degree of [tex]\(5\)[/tex] (since the exponent of [tex]\(y\)[/tex] is 5).
- [tex]\(-5x y^3\)[/tex] has a degree of [tex]\(4\)[/tex] (sum of exponents: [tex]\(1 + 3\)[/tex]).
- [tex]\(7x^2 y^2\)[/tex] has a degree of [tex]\(4\)[/tex] (sum of exponents: [tex]\(2 + 2\)[/tex]).
- [tex]\(-x^3 y\)[/tex] has a degree of [tex]\(4\)[/tex] (sum of exponents: [tex]\(3 + 1\)[/tex]).
- [tex]\(-6x^4\)[/tex] has a degree of [tex]\(4\)[/tex] (since the exponent of [tex]\(x\)[/tex] is 4).
The highest degree term is [tex]\(8y^6\)[/tex] with a degree of [tex]\(6\)[/tex]. Hence, the degree of this polynomial is [tex]\(6\)[/tex].
4. [tex]\(-6x y^5 + 5x^2 y^3 - x^3 y^2 + 2x^2 y^3 - 3x y^5\)[/tex]
- [tex]\(-6x y^5\)[/tex] has a degree of [tex]\(6\)[/tex] (sum of exponents: [tex]\(1 + 5\)[/tex]).
- [tex]\(5x^2 y^3\)[/tex] has a degree of [tex]\(5\)[/tex] (sum of exponents: [tex]\(2 + 3\)[/tex]).
- [tex]\(-x^3 y^2\)[/tex] has a degree of [tex]\(5\)[/tex] (sum of exponents: [tex]\(3 + 2\)[/tex]).
- [tex]\(2x^2 y^3\)[/tex] has a degree of [tex]\(5\)[/tex] (sum of exponents: [tex]\(2 + 3\)[/tex]).
- [tex]\(-3x y^5\)[/tex] has a degree of [tex]\(6\)[/tex] (sum of exponents: [tex]\(1 + 5\)[/tex]).
The highest degree term is [tex]\(-6x y^5\)[/tex] or [tex]\(-3x y^5\)[/tex], both with a degree of [tex]\(6\)[/tex]. Hence, the degree of this polynomial is [tex]\(6\)[/tex].
From our examination, the only polynomial with a degree of [tex]\(5\)[/tex] is:
[tex]\[ 2x y^4 + 4x^2 y^3 - 6x^3 y^2 - 7x^4 \][/tex]
Thus, the algebraic expression which is a polynomial with a degree of [tex]\(5\)[/tex] is:
[tex]\[ 2x y^4 + 4x^2 y^3 - 6x^3 y^2 - 7x^4 \][/tex]
We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.
