Skill: Classify polynomials by degree and number of terms

To classify a polynomial, determine two things:

1. Its degree: the greatest exponent of the variable after combining like terms.
2. Its number of terms: count the nonzero terms after simplifying.

Use this example:

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

Step 1: Combine like terms

The terms $4x^2$ and $-x^2$ are like terms:

$$4x^2-x^2=3x^2$$

So the polynomial becomes:

$$3x^2+2x+3$$

Step 2: Find the degree

Look at the exponents of $x$:

• $3x^2$ has exponent $2$
• $2x$ has exponent $1$
• $3$ is a constant, so its exponent is $0$

The greatest exponent is $2$, so the polynomial has degree $2$. A degree-$2$ polynomial is called quadratic.

Step 3: Count the terms

The simplified polynomial has three nonzero terms:

$$3x^2,\qquad 2x,\qquad 3$$

A polynomial with three terms is called a trinomial.

Therefore,

$$\boxed{4x^2+3-x^2+2x\text{ is a quadratic trinomial.}}$$

Remember: Always combine like terms before finding the degree or counting the terms.