polynomial function

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A polynomial of degree n is a function of the form f(x) = anx n + an−1x n−1 + . . . + a2x 2 + a1x + a0 where the a’s are real numbers (sometimes called the coefficients of the polynomial). Although this general formula might look quite complicated, particular examples are much simpler. For example, f(x) = 4x 3 − 3x 2 + 2 is a polynomial of degree 3, as 3 is the highest power of x in the formula. This is called a cubic polynomial, or just a cubic. And f(x) = x 7 − 4x 5 + 1 is a polynomial of degree 7, as 7 is the highest power of x. Notice here that we don’t need every power of x up to 7: we need to know only the highest power of x to find out the degree. An example of a kind you may be familiar with is f(x) = 4x 2 − 2x − 4 which is a polynomial of degree 2, as 2 is the highest power of x. This is called a quadratic. Functions containing other operations, such as square roots, are not polynomials. For example, f(x) = 4x 3 + √ x − 1 is not a polynomial as it contains a square root. And f(x) = 5x 4 − 2x 2 + 3/x is not a polynomial as it contains a ‘divide by x’.

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r

A polynomial of degree n is a function of the form f(x) = anx n + an−1x n−1 + . . . + a2x 2 + a1x + a0 where the a’s are real numbers (sometimes called the coefficients of the polynomial). Although this general formula might look quite complicated, particular examples are much simpler. For example, f(x) = 4x 3 − 3x 2 + 2 is a polynomial of degree 3, as 3 is the highest power of x in the formula. This is called a cubic polynomial, or just a cubic. And f(x) = x 7 − 4x 5 + 1 is a polynomial of degree 7, as 7 is the highest power of x. Notice here that we don’t need every power of x up to 7: we need to know only the highest power of x to find out the degree. An example of a kind you may be familiar with is f(x) = 4x 2 − 2x − 4 which is a polynomial of degree 2, as 2 is the highest power of x. This is called a quadratic. Functions containing other operations, such as square roots, are not polynomials. For example, f(x) = 4x 3 + √ x − 1 is not a polynomial as it contains a square root. And f(x) = 5x 4 − 2x 2 + 3/x is not a polynomial as it contains a ‘divide by x’. K

n algebra, the polynomial remainder theorem or little Bézout's theorem is an application of Euclidean division of polynomials. It states that the remainder of the division of a polynomial by a linear polynomial is equal to. In particular, is a divisor of if and only if.

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A polynomial function is a function such as a quadratic, a cubic, a quartic, and so on, involving only non-negative integer powers of x. We can give a general defintion of a polynomial, and define its degree.

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Polynomial comes from poly- (meaning "many") and -nomial (in this case meaning "term") ... so it says "many terms"
A polynomial can have:

constants (like 3, −20, or ½)
variables (like x and y)
exponents (like the 2 in y2), but only 0, 1, 2, 3, ... etc are allowed
that can be combined using addition, subtraction, multiplication and division ...

... except ...

... not division by a variable (so something like 2/x is right out)
So:

A polynomial can have constants, variables and exponents,
but never division by a variable.

Synthetic division is a shorthand, or shortcut, method of polynomial division in the special case of dividing by a linear factor -- and it only works in this case. Synthetic division is generally used, however, not for dividing out factors but for finding zeroes (or roots) of polynomials. More about this later.

If you are given, say, the polynomial equation y = x2 + 5x + 6, you can factor the polynomial as y = (x + 3)(x + 2). Then you can find the zeroes of y by setting each factor equal to zero and solving. You will find that x = –2 and x = –3 are the two zeroes of y.

You can, however, also work backwards from the zeroes to find the originating polynomial. For instance, if you are given that x = –2 and x = –3 are the zeroes of a quadratic, then you know that x + 2 = 0, so x + 2 is a factor, and x + 3 = 0, so x + 3 is a factor. Therefore, you know that the quadratic must be of the form y = a(x + 3)(x + 2).