Question

which expression is a trinomial of degree 1?

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

\\(y^2\\)

\\(6 + x - y\\)

\\(3x - y\\)

Explanation:

🆕 New Concept Discovered: Classifying Polynomials
Identifying polynomials by their number of terms and degree.

Step 1: Understand "Trinomial"

A trinomial is an algebraic expression containing exactly three terms separated by addition or subtraction signs. Let's count the terms in each option:

• \( x^3 + x^2 - x \): Three terms (\( x^3 \), \( x^2 \), and \( -x \)) \(

ightarrow\) Trinomial

• \( y^2 \): One term \(

ightarrow\) Monomial

• \( 6 + x - y \): Three terms (\( 6 \), \( x \), and \( -y \)) \(

ightarrow\) Trinomial

• \( 3x - y \): Two terms (\( 3x \), and \( -y \)) \(

ightarrow\) Binomial

This leaves us with \( x^3 + x^2 - x \) and \( 6 + x - y \) as our trinomial candidates.

Step 2: Understand "Degree 1"

The degree of a polynomial is the highest exponent of its variable(s) in any single term.

• For \( x^3 + x^2 - x \): The highest exponent is \( 3 \) (from the \( x^3 \) term). This is a degree 3 (cubic) trinomial.
• For \( 6 + x - y \): The variables \( x \) and \( y \) both have an implicit exponent of \( 1 \) (since \( x = x^1 \) and \( y = y^1 \)). The highest exponent is \( 1 \). This is a degree 1 (linear) trinomial.

Therefore, \( 6 + x - y \) is a trinomial of degree 1.

Answer:

\( 6 + x - y \)