Question

what is the degree of each polynomial?

\\((x^2 - y)^2\\) degree

\\(8x^4 - 5x^7 + 4x^5\\) degree

\\(\frac{x^2y^3}{3} + 2x^3 + 8x^3\\) degree

\\(x^4 + 2x^3 - 6x^4 - 17\\) degree

\\(-x^2 + 7x - 4x\\) degree

\\(x^7 + y^8 + x^7 - y^8\\) degree

Explanation:

🆕 New Concept Discovered: Degree of a Polynomial
The highest exponent sum in any single term after simplifying.

Step 1: Analyze \( (x^2 - y)^2 \)

Expand the expression to find the individual terms:
\[ (x^2 - y)^2 = (x^2)^2 - 2(x^2)(y) + y^2 = x^4 - 2x^2y + y^2 \]

Now, find the degree of each term (the sum of the exponents of the variables in that term):

• \( x^4 \): degree is \( 4 \)
• \( -2x^2y \): degree is \( 2 + 1 = 3 \)
• \( y^2 \): degree is \( 2 \)

The highest degree among these terms is \( 4 \).

Step 2: Analyze \( 8x^4 - 5x^7 + 4x^5 \)

Identify the degree of each term in this single-variable polynomial:

• \( 8x^4 \): degree is \( 4 \)
• \( -5x^7 \): degree is \( 7 \)
• \( 4x^5 \): degree is \( 5 \)

The highest exponent is \( 7 \).

Step 3: Analyze \( \frac{x^2y^3}{3} + 2x^3 + 8x^3 \)

First, simplify the expression by combining like terms:
\[ \frac{x^2y^3}{3} + 10x^3 \]

Now, find the degree of each term:

• \( \frac{x^2y^3}{3} \): degree is the sum of the exponents of \( x \) and \( y \), which is \( 2 + 3 = 5 \)
• \( 10x^3 \): degree is \( 3 \)

The highest degree is \( 5 \).

Step 4: Analyze \( x^4 + 2x^3 - 6x^4 - 17 \)

First, simplify the expression by combining the like terms \( x^4 \) and \( -6x^4 \):
\[ (1 - 6)x^4 + 2x^3 - 17 = -5x^4 + 2x^3 - 17 \]

Now, find the degree of each term:

• \( -5x^4 \): degree is \( 4 \)
• \( 2x^3 \): degree is \( 3 \)
• \( -17 \): degree is \( 0 \)

The highest degree is \( 4 \).

Step 5: Analyze \( -x^2 + 7x - 4x \)

First, simplify by combining the like terms \( 7x \) and \( -4x \):
\[ -x^2 + 3x \]

Now, find the degree of each term:

• \( -x^2 \): degree is \( 2 \)
• \( 3x \): degree is \( 1 \)

The highest degree is \( 2 \).

Step 6: Analyze \( x^7 + y^8 + x^7 - y^8 \)

First, simplify by combining like terms:

• The \( y^8 \) and \( -y^8 \) terms cancel each other out: \( y^8 - y^8 = 0 \)
• Combine the \( x^7 \) terms: \( x^7 + x^7 = 2x^7 \)

The simplified polynomial is:
\[ 2x^7 \]

The degree of this single term is \( 7 \).

Answer:

• \( (x^2 - y)^2 \) $

ightarrow$ Degree: 4

• \( 8x^4 - 5x^7 + 4x^5 \) $

ightarrow$ Degree: 7

• \( \frac{x^2y^3}{3} + 2x^3 + 8x^3 \) $

ightarrow$ Degree: 5

• \( x^4 + 2x^3 - 6x^4 - 17 \) $

ightarrow$ Degree: 4

• \( -x^2 + 7x - 4x \) $

ightarrow$ Degree: 2

• \( x^7 + y^8 + x^7 - y^8 \) $

ightarrow$ Degree: 7