Polynomial

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When you see letters mixed with numbers and arithmetic, like 7x⁴-3x³+19x²-8x+197, there is a good chance that it is a polynomial. Mathematicians, scientists, and engineers use polynomials to solve problems. Polynomials are taught in algebra, which is a gateway course to all technical subjects.

In algebra, when you see letters, numbers, and arithmetic, the understanding is that the letters stand either for numbers you do not know (and may want to find) or for numbers that change during the course of the problem, variables, such as time. If the only arithmetic is addition, subtraction, multiplication, and whole number exponents, then that mixture of letters, numbers, and arithmetic is called a polynomial. If harder operations are used, such as division or square roots, then what you have is an "algebraic expression". A polynomial is an easy-to-use algebraic expression.

Polynomials are often used to form polynomial equations, such as the equation 7x⁴-3x³+19x²-8x+197 = 0, or polynomial functions, such as f(x) = 7x⁴-3x³+19x²-8x+197.

Talking About Polynomials[change | change source]

The most important thing to remember about a polynomial is "multiplication is understood". That means, for example, that 2x means two times x, or twice x. If x is 7, then 2x is 14.

The parts of a polynomial separated by plus or minus signs are called "terms". The plus sign or minus sign are part of the term. Thus, in the polynomial x2+2x-7, the terms are:

7x⁴
-3x³
+19x²
-8x
+197

If a polynomial has only one term, it is called a "monomial". 5x3 is a monomial. The multiplier out in front is called a "coefficient", the letter is called an "unknown" or a "variable", and the raised number after the x is called an exponent. On a calculator, and on some computers, instead of putting an exponent above and to the right of the x the symbol ^ is used, so that the monomial above could be written 5x^3.

A polynomial with exactly three terms is called a "trinomial".

A polynomial with exactly two terms is called a "binomial".

A term with no letters in it is called a "constant term".

A term with one letter but no exponent is called a "first degree term" or a "linear term".

A term with one letter which has exponent 2 is called a "second degree term" or "quadratic term". A "quadratic equation" is an equation in which the largest exponent on any term is 2.

A term with one letter which has exponent 3 is called a "third degree term" or "cubic term." A "cubic equation" is an equation in which the largest exponent on any term is 3.

A term with one letter which has exponent 4 is called a "fourth degree term" or "quartic term." A "quartic equation" is an equation in which the largest exponent on any term is 4.

A term with one letter which has exponent 5 is called a "fifth degree term" or "quintic term." A "quintic equation" is an equation in which the largest exponent on any term is 5.

A term with one letter which has exponent 6 is called a "sixth degree term" or "sextic term." A "sextic equation" is an equation in which the largest exponent on any term is 6.