Question

consider the following polynomial:
\\-9 + x^4 - 5x + 2x^3 + 5x\\
this polynomial can be classified as

• fifth degree
• fourth degree
• monomial
• binomial
• trinomial

Explanation:

🆕 New Concept Discovered: Classifying Polynomials
Identifying degree and number of terms

Step 1: Simplify the expression

Combine like terms in the given polynomial:

$$ -9 + x^4 - 5x + 2x^3 + 5x $$

The terms \( -5x \) and \( 5x \) are like terms and cancel each other out:

$$ -5x + 5x = 0 $$

The simplified polynomial is:

$$ x^4 + 2x^3 - 9 $$

Step 2: Determine the degree

The degree of a polynomial is the highest exponent of the variable in its simplified form.

In \( x^4 + 2x^3 - 9 \), the highest exponent is \( 4 \). Therefore, it is a fourth degree polynomial.

Step 3: Determine the number of terms

Count the number of terms in the simplified polynomial:

1. \( x^4 \)
2. \( 2x^3 \)
3. \( -9 \)

Since there are three terms, it is classified as a trinomial.

Step 4: Match with the given options

Looking at the options provided:

• fifth degree
• fourth degree (Correct)
• monomial
• binomial
• trinomial (Correct)

Answer:

The polynomial can be classified as:

• fourth degree
• trinomial