Question

1. write in standard form and classify by degree and number of terms: $5x^2 - 3x^4 + 6x - x^3 + 6$

Explanation:

Step1: Rearrange terms by degree

To write a polynomial in standard form, we arrange the terms in descending order of their degrees. The degree of a term is the exponent of the variable.

• The term \(-3x^4\) has degree 4.
• The term \(-x^3\) has degree 3.
• The term \(5x^2\) has degree 2.
• The term \(6x\) has degree 1.
• The term \(6\) (or \(6x^0\)) has degree 0.

So, rearranging the polynomial \(5x^2 - 3x^4 + 6x - x^3 + 6\) in descending order of degrees, we get:
\(-3x^4 - x^3 + 5x^2 + 6x + 6\)

Step2: Classify by degree

The degree of a polynomial is the highest degree of its terms. In the polynomial \(-3x^4 - x^3 + 5x^2 + 6x + 6\), the highest degree term is \(-3x^4\) with degree 4. A polynomial of degree 4 is called a quartic polynomial.

Step3: Classify by number of terms

We count the number of terms in the polynomial. The terms are \(-3x^4\), \(-x^3\), \(5x^2\), \(6x\), and \(6\). So, there are 5 terms. A polynomial with 5 terms is called a quintic polynomial? Wait, no. Wait, the number of terms: let's list them again. Wait, \(-3x^4\) (1), \(-x^3\) (2), \(5x^2\) (3), \(6x\) (4), \(6\) (5). So 5 terms. A polynomial with 5 terms is a polynomial with 5 monomials, so we can call it a five - term polynomial, or more specifically, a quintinomial? Wait, actually, the general names are based on the number of terms: 1 - monomial, 2 - binomial, 3 - trinomial, 4 - quadrinomial, 5 - quintinomial. But sometimes, for polynomials with more than 3 terms, we just say a polynomial with \(n\) terms. But in terms of classification by number of terms, we can say it's a polynomial with 5 terms (or a quintinomial). And by degree, it's a quartic polynomial (degree 4).

Answer:

Standard form: \(\boldsymbol{-3x^4 - x^3 + 5x^2 + 6x + 6}\)

Classification by degree: Quartic (degree 4)

Classification by number of terms: Polynomial with 5 terms (or quintinomial)