Lainey Goodwin

2022-01-29

What is a polynomial with 4 terms?

Expert

Explanation:
$a{x}^{3}+b{x}^{2}+cx+d$ is a quadrinomial and a cubic.
$a{x}^{5}+b{x}^{2}+cx+d$ is quadrinomial but a quintic (the term of highest degree has degree 5).
$a{x}^{3}+cx+d$ is a cubic but not a quadrinomial.
$a{x}^{3}+b{x}^{2}+cx+\frac{d}{x}$ is a quadrinomial but not a polynomial.

Deegan Mullen

Expert

A polynomial with 1 term is called a monomial.
Examples: $3{x}^{2},5x,7.$
A polynomial with 2 terms is called a binomial.
Examples: $x+y,5{x}^{3}+7,4{x}^{7}+23{x}^{3}.$
A polynomial with 3 terms is called a trinomial.
Examples: $x+y+z,{x}^{2}+5x-7,{x}^{6}-7{y}^{3}+12x.$
So far as I know there is no standard term for a polynomial with 4 terms.
However, the number of terms in a polynomial is not very important.
The two important things about a polynomial are the number of variables.
For example, this polynomial ${x}^{2}+{y}^{2}-24$ has two variables x, and y; but this polynomial $7{x}^{2}-3x+8$ has only one variable.
The other important thing about a polynomial is its degree, which in the case of a polynomial of one variable is the largest exponent, so for example the polynomial ${x}^{3}-7{x}^{2}+11x-17$ has four terms and is of degree 3. In case the polynomial has more than one variable, the degree of each term is the sum of the exponents of the variables in that term and the degree of the polynomial is the number which is the degree of that term which has highest degree. So, for example, in the polynomial $4{x}^{2}{y}^{3}+7xy-5{x}^{4}+6$, the degree of the first term is 2+3=5, the degree of the second term is 1+1=2, the degree of the third term is 4 and the degree of the constant term is 0, so the degree of the entire polynomial is the largest of those, namely 5.
Polynomials of degree 1 are called linear, polynomials degree 2 are called quadratics, polynomials of degree 3 are called cubics, polynomials of degree 4 are called quartics, and polynomials of degree 5 are called quintics.

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