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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]
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