Question

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

Answer 1

A polynomial n degree in standard form is,
$p(x) = {a_n}{x^n} + {a_{n - 1}}{x^{n - 1}} + ... + {a_2}{x^2} + {a_1}x + {a_0}$,
Which have $n + 1$ terms (from ${a_0}$ to ${a_n}$)
In standard form, the terms of a polynomial are arranged in decreasing degree, terms of a polynomial are the subcomponents of the polynomial connected by to other components by only addition or subtraction. The degree of each term in a polynomial in two variables is the sum of the exponents in each term.
So here we have written the polynomial according to the degree with highest to smallest degree.

Complete step-by-step answer:
A polynomial n degree in standard form is,
$p(x) = {a_n}{x^n} + {a_{n - 1}}{x^{n - 1}} + ... + {a_2}{x^2} + {a_1}x + {a_0}$,
Which have$n + 1$terms (from ${a_0}$ to ${a_n}$)
Polynomials in two variables are algebraic expressions consisting of terms in the form. The degree of each term in a polynomial in two variables is the sum of the exponents in each term and the degree of the polynomial is the largest such sum.
To classify a polynomial by degree, you look at the highest exponent, or degree.
Polynomial with degree 1 is called linear,
Polynomial with degree 2 is called quadratic,
Polynomial with degree 3 is called cubic,
Polynomial with degree 4 is called quadratic,
To classify a polynomial by the number of terms, count how many terms are in the polynomial,
Polynomial with 1 term is called monomial,
Polynomial with 2 terms is called binomial,
Polynomial with 3 terms is called trinomial,
Polynomial with 4 terms is called a polynomial.
To write a polynomial in standard form, you write starting with the term with the highest degree, or exponent and then in decreasing order,
Here $4{x^4}{y^5}$, is the term with the highest degree. The expression ${x^4}{y^5}$ has an exponent of 4 on the $x$ and a written exponent of 5 on the $y$, so this term is to the third degree (4+5). Notice that you add the two degrees together because it has two variables, so the degree is 9,
And next term with the highest degree is $- 3{x^4}{y^2}$. The expression ${x^4}{y^2}$ has an exponent of 4 on the $x$ and a written exponent of 2 on the$y$, so this term is to the third degree (4+2). Notice that you add the two degrees together because it has two variables, so the degree is 6,
And the last term will be $10{x^2}$, with degree 2,
So the standard term will be $4{x^4}{y^5} - 3{x^4}{y^2} + 10{x^2}$.

$\therefore$ The standard form of $- 3{x^4}{y^2} + 4{x^4}{y^5} + 10{x^2}$ will be $4{x^4}{y^5} - 3{x^4}{y^2} + 10{x^2}$.

Note:
To write a polynomial is in standard form. In order to write any polynomial in standard form, you look at the degree of each term. You then write each term in order of degree, from highest to lowest, left to right.
Let’s look at an example.
Write the expression $3x - 8 + 4{x^5}$ in standard form.
First, look at the degrees for each term in the expression.
$3x$ has a degree of 1
8 has a degree of 0
$4{x^5}$ has a degree of 5
Next, write this trinomial in order by degree, highest to lowest
$4{x^5} + 3x - 8$,
The answer is $4{x^5} + 3x - 8$.
The degree of a polynomial is the same as the degree of the highest term, so this expression is called a fifth degree trinomial.