Question

What type of polynomial is ${x^2} - 2x + 1$ ?

Answer 1

Hint:A polynomial is an expression that can be built from constants and symbols called variables or indeterminants by means of addition, multiplication and exponentiation to a non-negative integer power. degree of a polynomial is the highest of the degrees of the polynomial's monomials (individual terms) with non-zero coefficients. For a univariate polynomial, the degree of the polynomial is simply the highest exponent occurring in the polynomial.

Complete step by step answer:
A polynomial in a single indeterminate x can always be written (or rewritten) in the form
${a_n}{x^n} + {a_{n - 1}}{x^{n - 1}} + .......{a_2}{x^2} + {a_1}x + {a_0}$ where ${a_0},{a_1}......,{a_n}$ are constants and $x$ is the indeterminate. The word “indeterminate” means that $x$ represents no particular value , although any value may be substituted for it.

In mathematics, the degree of a polynomial is the highest of the degrees of the polynomial's monomials (individual terms) with non-zero coefficients. The degree of a term is the sum of the exponents of the variables that appear in it, and thus is a non-negative integer. For a univariate polynomial, the degree of the polynomial is simply the highest exponent occurring in the polynomial.

The following names are assigned to polynomials according to their degree:
Degree 0 – non-zero constant
Degree 1 – linear
Degree 2 – quadratic
Degree 3 – cubic
Degree 4 – quartic (or, if all terms have even degree, biquadratic)
Degree 5 – quintic
Degree 6 – sextic (or, less commonly, hexic)
Degree 7 – septic (or, less commonly, heptic)

For higher degrees, names have sometimes been proposed] but they are rarely used:
Degree 8 – octic
Degree 9 – nonic
Degree 10 – decic
Here in the question the polynomial is ${x^2} - 2x + 1$ . This polynomial only contains one variable that is $x$. Thus it is a univariate polynomial. Hence the degree of the polynomial is simply the highest exponent occurring in the polynomial. Here the highest exponent is $2$ , therefore the degree of this polynomial is $2$ .

Hence this is a quadratic polynomial in $x$.

Note:Take care that the polynomial is in one variable or more than one variable. If it is in one variable, the degree is simply the highest exponent occurring in the polynomial. If it is in more than one variable, the degree is the highest sum of exponents occurring in the polynomial.