POLYNOMIALS
Introduction:
An algebraic expression is a collection of variables and real numbers. The most common type of algebraic expression is the polynomial. The 'polly' in 'polynomial' means 'many'. The term 'polynomial' refers to sums of many terms, but the term is used to refer to anything from one term to the sum of a zillion terms.
Terminology
To create a polynomial, one takes some terms and adds (and subtracts) them together. Here is a typical polynomial:
Notice the exponents (that is, the powers) on each of the three terms. The first term has an exponent of 2; the second term has an "understood" exponent of 1 (which customarily is not included); and the last term doesn't have any variable at all, so exponents aren't an issue. Because there is no variable in this last term, it's value never changes, so it is called the "constant" term
The first term in the polynomial, when that polynomial is written in descending order, is also the term with the biggest exponent, and is called the "leading" term.
Polynomial Names
The "poly-" prefix in "polynomial" means "many", from the Greek language. (The "-nomial" part might come from the Latin for "named", but this isn't certain.) I suppose, technically, the term "polynomial" should refer only to sums of many terms, but "polynomial" is used to refer to anything from one term to the sum of a zillion terms.
However, the shorter polynomials do have their own names, according to their number of terms
monomial: a one-term polynomial, such as 2x or 4x2 ("mono-" meaning "one")
binomial: a two-term polynomial, such as 2x + y or x2 – 4 ("bi-" meaning "two")
trinomial: a three-term polynomial, such as 2x + y + z or x4 + 4x2 – 4 ("tri-" meaning "three")
Polynomials are also sometimes named for their degree
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• linear: a first-degree polynomial, such as 6x or –x + 2 (because it graphs as a straight line)
• quadratic: a second-degree polynomial, such as 4x2, x2 – 9, or ax2 + bx + c (from the Latin "quadraticus", meaning "made square")
• cubic: a third-degree polynomial, such as –6x3 or x3 – 27 (because the variable in the leading term is cubed, and the suffix "-ic" in English means "pertaining to")
• quartic: a fourth-degree polynomial, such as x4 or 2x4 – 3x2 + 9 (from the Latic "quartus", meaning "fourth")
• quintic: a fifth-degree polynomial, such as 2x5 or x5 – 4x3 – x + 7 (from the Latic "quintus", meaning "fifth")
Determine the degree of the polynomial, and list the values of the leading coefficient and the constant term, if any, of the following polynomial: 6x2 + 7x4 + x
This polynomial has three terms: a second-degree term, a fourth-degree term, and a first-degree term. There is no constant term
The three terms are not written in descending order, I notice. The 6x2, while written first, is not the "leading" term, because it does not have the highest degree. The highest-degree term is the 7x4, so this is a degree-four polynomial. Also, this term, though not listed first, is the actual leading term; its coefficient is 7.
examples answer:
degree: 4
leading coefficient: 7
constant: none