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The general polynomial can be a function of several variables in any degree. The polynomial in one variable, in its normal form, is written as:

f(x) = a_nxⁿ + a_n-1x^n-1 + a _n-2x^n-2 + ... + a₂x² + a₁x + a₀

where a_n is a complex number, x ranges over the set of complex numbers, and n is any positive integer.

The general polynomial of degree n in x is said to be an nth degree polynomial in the variable x. In general, n is a positive integer and there are n+1 constants in the normal form.

Classes of polynomial functions in one variable include the linear and quadratic functions, of degree one and two, respectively. A function is linear if the coefficient, a₁, is non-zero and the coefficients a₂ through a_n are zero. A function is quadratic if a₂ is non-zero and coefficients a₃ through a_n are zero. Similar statements apply to higher degree polynomials of.

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When working with polynomial functions, you need to understand what exponents stand for and how they are to be treated. Say, for example, you have the 4th degree function:

f(x) = 4x⁴ - 3x..

This function has two exponents, 4 and 1, even though the latter isn't written out. Exponents generally show how many times the base of the exponent -- in this case x -- appears as a factor. In the expression, xⁿ, n is the power of x. In x³, for instance, 3 is the power of x and says that x is a factor three times. So x³ is the same as x times x times x, or xxx.

If you multiply x³ by x³, you get the factor, x, six times, or:

x times x times x times x times x times x,

which is:

xxxxxx.

But this is the same as x⁶. And so you have the result:

(x³)( x³) = x⁶.

You get similar results using any number of such factors, so you can avoid repeating the factors and simply write the general rule as:

(x^a)( x^b) = x^a+b.

In other words, when you multiply similar factors having exponents, you add the exponents. This is shorthand. If the factor base is different in each factor, you cannot and must not add their exponents. For instance, from

(x^a)( y^b)

you mustn't try to get

(xy)^a+b.

This works only if x equals y.

The same ideas apply if you divide, except that now you add a negative number in the exponent, which is just subtracting the positive number. For example, let's divide x³ by x². This quotient is:

x³/x² = (xxx/xx)

In division you can cancel like factors.  So you end up with:

x,

or:

x^3-2 = x¹ = x.

Since x^-1 stands for x, you can also write:

x^3+(-2) = x^3-2 = x¹ = x.

And the general rule for division becomes:

x^a/x^b = x^a-b

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A polynomial is really a string of simpler terms added together, where the terms of the addition are called monomials -- the 'mono' prefix meaning 'one.'  Referring to the general polynomial, here, you can see that the various terms -- the monomials -- have one of the forms:

a, or a_nxⁿ.

For example, you could have the constant:

7 + 3i

where i is the imaginary part of complex numbers. Or you could have the monomial of power 9 in x:

5x⁹.

Note that 5x⁹ is the sum of five of the x⁹s:

5x⁹ = + x⁹ + x⁹ + x⁹ + x⁹ + x⁹.

When you add monomials, you can be dealing with variables of different degree. So you must be sure to merge only monomials of the same degree. For instance, 3x⁵ + x² isn't equal to 4x⁷. They can't be simplified to one term.

So now let's add two polynomials. For instance, let's add

(5x⁴ + 3x² + 2x)

and

(3x⁵ - 7x⁴ + x²).

Rearranging and combining the appropriate monomials in the polynomials, we get:

3x⁵ + 5x⁴ - 7x⁴ + 3x² + x² + 2x = 3x⁵ -2x⁴ + 4x² + 2x.

Subtracting a polynomial is the same as adding the negatives of the polynomial. So let's subtract

(4x⁴ - 3x² + 2x)

from

(5x⁴- 3x³ + x² +3)

First line up vertically the monomials to be merged. In other words, arrange the terms in descending order, just as in arithmetic, one polynomial below the other, and subtract only corresponding terms.

Notice that I created a direct match between the polynomials by bringing out the three missing terms -- using zeros. So:

(5x⁴- 3x³ + x² + 0x + 3) - (4x⁴ - 3x³ - 0x² + 2x + 0) = x⁴ + x² - 2x + 3

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Multiplication and division of polynomials follow similar practices of multiplication and division of numbers, which is what you'd expect, inasmuch as we're dealing with numbers.

Let's multiply (5x⁴- 3x³ + x² + 1) by (x + 2). Playing by the algebraic rules of multiplication, the product of the two polynomials is the sum of the products of the monomials. This is the distributive rule:

a(b + c) = ab + bc.

So we get, respectively, for x and 2:

5xx⁴- 3xx³ + xx² + 1

or

5x⁵- 3x⁴ + x³ + x

and

10x⁴ - 6x³ + 2x² + 2

Bracketing the terms with common exponents we can write:

5x⁵ + (10x⁴ - 3x⁴) + (x³ - 6x³) + 2x² + x + 2,

or

5x⁵ + 7x⁴ - 5x³ + 2x² + x + 2.

As an example, let's divide (5x³ - 6x²) by x². That is, we wish to find the quotient:

One approach is to rewrite the denominator as x^--2 and multiply, using the distributive rule of multiplication over the numerator, as follows:

x-²(5x³ - 6x²) = x^--25x³ - x^--26x²

Adding exponents in the two terms we have the quotient in the form:

5x¹ - 6x⁰ = 5x - 6

since

x⁰ = 1.

Any quantity raised to the zero power is 1.

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