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Sagot :
Let's find the degree of each given expression step-by-step.
### Expression 1: [tex]\((x^2 - y)^2\)[/tex]
To find the degree, we first expand the expression:
[tex]\[ (x^2 - y)^2 = (x^2 - y) \cdot (x^2 - y) = x^4 - 2x^2y + y^2 \][/tex]
- The degree of [tex]\(x^4\)[/tex] is 4.
- The degree of [tex]\(-2x^2y\)[/tex] is [tex]\(2 + 1 = 3\)[/tex].
- The degree of [tex]\(y^2\)[/tex] does not involve [tex]\(x\)[/tex], so it is irrelevant for the degree in terms of [tex]\(x\)[/tex].
The term with the highest degree in [tex]\(x\)[/tex] is [tex]\(x^4\)[/tex], so the degree is:
[tex]\[ \boxed{4} \][/tex]
### Expression 2: [tex]\(8x^4 - 5x^7 + 4x^5\)[/tex]
Identify the degrees of each term:
- [tex]\(8x^4\)[/tex] has degree 4.
- [tex]\(-5x^7\)[/tex] has degree 7.
- [tex]\(4x^5\)[/tex] has degree 5.
The term with the highest degree is [tex]\(-5x^7\)[/tex], so the degree is:
[tex]\[ \boxed{7} \][/tex]
### Expression 3: [tex]\(\frac{x^2 y^3}{3} + 2x^3 + 8x^3\)[/tex]
Combine like terms first:
[tex]\[ \frac{x^2 y^3}{3} + 2x^3 + 8x^3 = \frac{x^2 y^3}{3} + 10x^3 \][/tex]
Identify the degrees of each term:
- [tex]\(\frac{x^2 y^3}{3}\)[/tex] has degree 2 for [tex]\(x\)[/tex].
- [tex]\(10x^3\)[/tex] has degree 3.
The term with the highest degree in [tex]\(x\)[/tex] is [tex]\(10x^3\)[/tex], so the degree is:
[tex]\[ \boxed{3} \][/tex]
### Expression 4: [tex]\(x^4 + 2x^3 - 6x^4 - 17\)[/tex]
Combine like terms:
[tex]\[ x^4 + 2x^3 - 6x^4 - 17 = -5x^4 + 2x^3 - 17 \][/tex]
Identify the degrees of each term:
- [tex]\(-5x^4\)[/tex] has degree 4.
- [tex]\(2x^3\)[/tex] has degree 3.
- [tex]\(-17\)[/tex] does not involve [tex]\(x\)[/tex].
The term with the highest degree is [tex]\(-5x^4\)[/tex], so the degree is:
[tex]\[ \boxed{4} \][/tex]
### Expression 5: [tex]\(-x^2 + 7x - 4x\)[/tex]
Combine like terms:
[tex]\[ -x^2 + 7x - 4x = -x^2 + 3x \][/tex]
Identify the degrees of each term:
- [tex]\(-x^2\)[/tex] has degree 2.
- [tex]\(3x\)[/tex] has degree 1.
The term with the highest degree is [tex]\(-x^2\)[/tex], so the degree is:
[tex]\[ \boxed{2} \][/tex]
### Expression 6: [tex]\(x^7 + y^8 + x^7 - y^8\)[/tex]
Combine like terms:
[tex]\[ x^7 + y^8 + x^7 - y^8 = 2x^7 \][/tex]
Identify the degrees of each term:
- [tex]\(2x^7\)[/tex] has degree 7.
The term with the highest degree is [tex]\(2x^7\)[/tex], so the degree is:
[tex]\[ \boxed{7} \][/tex]
So the degrees of the expressions are:
- [tex]\((x^2 - y)^2: \boxed{4} \)[/tex]
- [tex]\(8x^4 - 5x^7 + 4x^5: \boxed{7} \)[/tex]
- [tex]\(\frac{x^2 y^3}{3} + 2x^3 + 8x^3: \boxed{3} \)[/tex]
- [tex]\(x^4 + 2x^3 - 6x^4 - 17: \boxed{4} \)[/tex]
- [tex]\(-x^2 + 7x - 4x: \boxed{2} \)[/tex]
- [tex]\(x^7 + y^8 + x^7 - y^8: \boxed{7} \)[/tex]
### Expression 1: [tex]\((x^2 - y)^2\)[/tex]
To find the degree, we first expand the expression:
[tex]\[ (x^2 - y)^2 = (x^2 - y) \cdot (x^2 - y) = x^4 - 2x^2y + y^2 \][/tex]
- The degree of [tex]\(x^4\)[/tex] is 4.
- The degree of [tex]\(-2x^2y\)[/tex] is [tex]\(2 + 1 = 3\)[/tex].
- The degree of [tex]\(y^2\)[/tex] does not involve [tex]\(x\)[/tex], so it is irrelevant for the degree in terms of [tex]\(x\)[/tex].
The term with the highest degree in [tex]\(x\)[/tex] is [tex]\(x^4\)[/tex], so the degree is:
[tex]\[ \boxed{4} \][/tex]
### Expression 2: [tex]\(8x^4 - 5x^7 + 4x^5\)[/tex]
Identify the degrees of each term:
- [tex]\(8x^4\)[/tex] has degree 4.
- [tex]\(-5x^7\)[/tex] has degree 7.
- [tex]\(4x^5\)[/tex] has degree 5.
The term with the highest degree is [tex]\(-5x^7\)[/tex], so the degree is:
[tex]\[ \boxed{7} \][/tex]
### Expression 3: [tex]\(\frac{x^2 y^3}{3} + 2x^3 + 8x^3\)[/tex]
Combine like terms first:
[tex]\[ \frac{x^2 y^3}{3} + 2x^3 + 8x^3 = \frac{x^2 y^3}{3} + 10x^3 \][/tex]
Identify the degrees of each term:
- [tex]\(\frac{x^2 y^3}{3}\)[/tex] has degree 2 for [tex]\(x\)[/tex].
- [tex]\(10x^3\)[/tex] has degree 3.
The term with the highest degree in [tex]\(x\)[/tex] is [tex]\(10x^3\)[/tex], so the degree is:
[tex]\[ \boxed{3} \][/tex]
### Expression 4: [tex]\(x^4 + 2x^3 - 6x^4 - 17\)[/tex]
Combine like terms:
[tex]\[ x^4 + 2x^3 - 6x^4 - 17 = -5x^4 + 2x^3 - 17 \][/tex]
Identify the degrees of each term:
- [tex]\(-5x^4\)[/tex] has degree 4.
- [tex]\(2x^3\)[/tex] has degree 3.
- [tex]\(-17\)[/tex] does not involve [tex]\(x\)[/tex].
The term with the highest degree is [tex]\(-5x^4\)[/tex], so the degree is:
[tex]\[ \boxed{4} \][/tex]
### Expression 5: [tex]\(-x^2 + 7x - 4x\)[/tex]
Combine like terms:
[tex]\[ -x^2 + 7x - 4x = -x^2 + 3x \][/tex]
Identify the degrees of each term:
- [tex]\(-x^2\)[/tex] has degree 2.
- [tex]\(3x\)[/tex] has degree 1.
The term with the highest degree is [tex]\(-x^2\)[/tex], so the degree is:
[tex]\[ \boxed{2} \][/tex]
### Expression 6: [tex]\(x^7 + y^8 + x^7 - y^8\)[/tex]
Combine like terms:
[tex]\[ x^7 + y^8 + x^7 - y^8 = 2x^7 \][/tex]
Identify the degrees of each term:
- [tex]\(2x^7\)[/tex] has degree 7.
The term with the highest degree is [tex]\(2x^7\)[/tex], so the degree is:
[tex]\[ \boxed{7} \][/tex]
So the degrees of the expressions are:
- [tex]\((x^2 - y)^2: \boxed{4} \)[/tex]
- [tex]\(8x^4 - 5x^7 + 4x^5: \boxed{7} \)[/tex]
- [tex]\(\frac{x^2 y^3}{3} + 2x^3 + 8x^3: \boxed{3} \)[/tex]
- [tex]\(x^4 + 2x^3 - 6x^4 - 17: \boxed{4} \)[/tex]
- [tex]\(-x^2 + 7x - 4x: \boxed{2} \)[/tex]
- [tex]\(x^7 + y^8 + x^7 - y^8: \boxed{7} \)[/tex]
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