Question

which terms can be combined in the polynomial \\(x^4 + 6x^2 - 5x + 2x^3 - 9x^2\\)?

constant
linear
quadratic
cubic

Explanation:

🆕 New Concept Discovered: Combining Like Terms in Polynomials
Grouping terms with the same variable and exponent.

Step 1: Identify the terms in the polynomial

We are given the polynomial:

$$ x^4 + 6x^2 - 5x + 2x^3 - 9x^2 $$

Let's list each term and identify its degree (exponent):

• \( x^4 \) is a quartic term (degree 4)
• \( 6x^2 \) is a quadratic term (degree 2)
• \( -5x \) is a linear term (degree 1)
• \( 2x^3 \) is a cubic term (degree 3)
• \( -9x^2 \) is a quadratic term (degree 2)

Step 2: Find the like terms

To combine terms in a polynomial, the terms must be "like terms," meaning they must have the exact same variable raised to the exact same exponent.

Looking at our list:

• There is only one quartic term: \( x^4 \)
• There is only one cubic term: \( 2x^3 \)
• There is only one linear term: \( -5x \)
• There are two quadratic terms: \( 6x^2 \) and \( -9x^2 \)

Since \( 6x^2 \) and \( -9x^2 \) both have the variable \( x \) raised to the power of 2, they are like terms and can be combined. Terms with a degree of 2 are called quadratic terms.

Answer:

quadratic