Question

Which algebraic expression is a polynomial with a degree of 5?

3x5 + 8x4y2 – 9x3y3 – 6y5
2xy4 + 4x2y3 – 6x3y2 – 7x4
8y6 + y5 – 5xy3 + 7x2y2 – x3y – 6x4
–6xy5 + 5x2y3 – x3y2 + 2x2y3 – 3xy5

Answer 1

The correct answer is: [B]:
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→ " 2xy⁴ + 4x²y³ – 6x³y² – 7x⁴ " .
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Note:
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This answer choice is a polynomial with a degree of "5" .
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Note that there are 2 (TWO) variables in this polynomial expression
(in this very "correct answer choice: [B]:
→ which are: "x" and "y" .
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Consider each "term" in the polynomial given in

{ this correct answer choice: [B] } :

→ " 2xy⁴ + 4x²y³ – 6x³y² – 7x⁴ " ;

1) " 2xy⁴ " ; the degree is "5" ;
→ since " 2xy⁴ = 2x¹y⁴ " ;
→ and from " x¹y⁴ " ; the degree is: " 4 + 1 = 5 " .

2) " 4x²y³ " ; the degree is "5" ;
→ since from: " x²y³ " ; the degree is: " 2 + 3 = 5 " .

3) " 6x³y² " ; the degree is: "5" ;
→ since from: " x³y² " ; the degree is: " 3 + 2 = 5 " .

4) " 7x⁴ " ; the degree is "4" ;
→ since "4" is the only exponent of the only variable in the expression.

Since "5" is the highest degree of all of the terms in the polynomial;
→ the degree of the polynomial is: "5" .
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Consider the other answer choices:
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Choice: [A]: " 3x⁵ + 8x⁴y² – 9x³y³ – 6y⁵ " ; is incorrect.

→ By looking at the entire expression, we can see that the highest degree is at least "5" {that is; from the terms: " 3x⁵ " and "6y⁵ " . However, the other terms show us that the highest degree is "6" ; which is greater than "5".

→ Note: " 8x⁴y² " ; 4 + 2 = 6;
" 9x³y³ " ; 3 + 3 = 6 ;
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Choice: [C]: " 8y⁶ + y⁵ – 5xy³ + 7x²y² – x³y – 6x⁴ " ; is incorrect.

→ Note: From the first term in the expression:

" 8y⁶ " ; → we can already tell that the degree of the expression is [at least] "6" ; which is greater than "5" .

{Note: "6" is the only variable assigned to the only exponent within this term of the expression.}.
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Choice: [D]: " –6xy⁵ + 5x²y³ – x³y² + 2x²y³ – 3xy⁵ " ; is incorrect.

→ Note: From the first term in the expression:
" –6xy⁵ " ; → we can already tell that the degree of the expression is [at least] "6" ; which is greater than "5" .

→ Note: " –6xy⁵ " = " –6x¹y⁵ " ;

→ from: " x¹y⁵ " ; → " 1 + 5 = "6" .
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Answer 2

The Degree is the largest exponent of that variable.

Let's get started :

`3x^5 + 8x^4y^2 - 9x^3 y^3 - 6y^5`

Here, the largest exponent is from
`8x^4y^2`. It's 4+2 = 6.

`8y^6 + y^5 - 5xy^3 + 7x^2y^2 - x^3y - 6x^4`

Here, the largest exponent is 6.

`-6xy^5 + 5x^2y^3 - x^3y^2 + 2x^2y^3 - 3xy^5`

Here, the largest exponent is also 6. x itself is considered as exponent 1. So 1+5 = 6.

At last, let's check the second option,

`2xy^4 + 4x^2y^3 - 6x^3y^2 - 7x^4`

Here the largest exponent is 5.

In short, the answer would be :

`\boxed {\boxed{2xy^4 + 4x^2y^3 - 6x^3y^2 - 7x^4}}`

Hope this helps !

Photon