Question

How do you write a polynomial in standard form, then classify it by degree and number of terms $ - 3{{\text{x}}^4}{y^2} + 4{{\text{x}}^4}{y^5} + 10{x^2}$?

Answer 1

Hint: To compose any polynomial in standard form, you take a gander at the degree of each term. You at that point compose each term arranged by degree, from highest to lowest, left to right. To find the degree, we need to find the highest component of the polynomial and to find the number of terms we just need to count the number of terms.

Complete step by step answer:
Given: $ - 3{{\text{x}}^4}{y^2} + 4{{\text{x}}^4}{y^5} + 10{x^2}$
We need to write the polynomial in standard form, then classify it by degree and number of terms.
Since the term ${{\text{x}}^4}{y^5}$ $\left( {4 + 5 = 9} \right)$is the highest term in the given polynomial, we will write in decreasing order ${{\text{x}}^4}{y^2} = 6$ terms then ${x^2} = 2$terms.
Thus, we get the given polynomial in standard form as:
$4{{\text{x}}^4}{y^5} - 3{{\text{x}}^4}{y^2} + 10{x^2}$
To classify a polynomial by degree, we need to look at the highest degree.
Since $9$ is the highest exponent $\left( {4{{\text{x}}^4}{y^5}} \right)$,the degree of the polynomial is $9$.
Thus, it is a nonic equation.
To classify a polynomial by the number of terms, count how many terms are there in the polynomial.
In this polynomial there are three terms$\left( { - 3{{\text{x}}^4}{y^2}} \right),\left( {4{{\text{x}}^4}{y^5}} \right),\left( {10{x^2}} \right)$.Therefore, it is a trinomial.

Hence, the polynomial in standard form is $4{{\text{x}}^4}{y^5} - 3{{\text{x}}^4}{y^2} + 10{x^2}$ with degree $9$ and number of terms three.

Note: Polynomials can be arranged dependent on:
a) Number of terms b) Degree of the polynomial.
Types of polynomials dependent on the number of terms
Monomial
Binomial
Trinomial
Types of Polynomials dependent on Degree
Linear polynomial
Quadratic polynomial