Question

classify the following polynomials by the highest power of each of its terms. combine any like terms first.

\\(-x^2 + x - x^2 + 1\\)

\\(x^2 + x + 2x^3 - x\\)

\\(4x + x + x - 2\\)

\\(3x^2 + 4 - 3x^2 - 1\\)

Explanation:

🆕 New Concept Discovered: Classifying Polynomials by Degree
Identifying a polynomial's type by its highest exponent after simplifying.

Step 1: Classify the first polynomial

Combine like terms for \( -x^2 + x - x^2 + 1 \):
\[ (-x^2 - x^2) + x + 1 = -2x^2 + x + 1 \]
The highest power (degree) of the variable is \( 2 \). A polynomial of degree \( 2 \) is called quadratic.

Step 2: Classify the second polynomial

Combine like terms for \( x^2 + x + 2x^2 - x \):
\[ (x^2 + 2x^2) + (x - x) = 3x^2 \]
The highest power (degree) of the variable is \( 2 \). A polynomial of degree \( 2 \) is called quadratic.

Step 3: Classify the third polynomial

Combine like terms for \( 4x + x + x - 2 \):
\[ (4x + x + x) - 2 = 6x - 2 \]
The highest power (degree) of the variable is \( 1 \) (since \( x = x^1 \)). A polynomial of degree \( 1 \) is called linear.

Step 4: Classify the fourth polynomial

Combine like terms for \( 3x^2 + 4 - 3x^2 - 1 \):
\[ (3x^2 - 3x^2) + (4 - 1) = 3 \]
The variable terms cancel out completely, leaving only a constant number. A polynomial of degree \( 0 \) is called constant.

Answer:

• \( -x^2 + x - x^2 + 1 \) \(

ightarrow\) quadratic

• \( x^2 + x + 2x^2 - x \) \(

ightarrow\) quadratic

• \( 4x + x + x - 2 \) \(

ightarrow\) linear

• \( 3x^2 + 4 - 3x^2 - 1 \) \(

ightarrow\) constant