Polynomial Functions

Polynomial Functions So far we have understood about linear functions and quadratic functions. Now we shall generalize these ideas. We call an expression of the form a n x n + a n − x n − + · · · + a is called a polynomial in the variable x , where a i ∈ R , i = , , , · · · , n . Here n is a non-negative integer.

When a n = , we say that the polynomial has degree n . The numbers a , a , . . .

, a n ∈ R are called the coefficients of the polynomial. The number a is called the constant term and a n is called the leading coefficient (when it is non-zero). It is clear that: (i) x − πx + x + x + . is a polynomial of degree .

(ii) ( x − )( x + )( x − √ π )( x + . ) is a polynomial of degree . (iii) ( x + x + )( x + x + )( x − x + ) is a polynomial of degree . One may substitute specific values for x , say x = c and obtain a n c n + a n − c n − + · · · + a c + a .

A function of the form P ( x ) = a n x n + a n − x n − + · · · + a is called a polynomial function which is defined from R to R . We shall treat polynomial and polynomial function as one and the same. A polynomial with degree is called a linear polynomial . A polynomial with degree is called a quadratic polynomial .

A cubic polynomial is one that has degree three. Likewise, degree and degree polynomials are called quartic and quintic polynomials respectively. Note that any constant a = is a polynomial of degree zero! Two polynomials f ( x ) = a n x n + a n − x n − + · · · + a , a n = and g ( x ) = b m x m + b m − x m − + · · · + b , b m = are equal if and only if f ( x ) = g ( x ) for all x ∈ R .

It can be proved that f ( x ) = g ( x ) if and only if n = m and a k = b k , k = , , , · · · n . Given two polynomials, one can form their sum and product . For example if P ( x ) = x + x − and Q ( x ) = x − x + x + x + , then P ( x ) + Q ( x ) = x + x + x − (by adding the corresponding coefficients of the like powers of x ) and P ( x ) Q ( x ) = x + x − x + x + x + x − x − by multiplying each term of P ( x ) by every term of Q ( x ) . It is easy to see that the degree of P ( x ) Q ( x ) is the sum of the degrees of P ( x ) and Q ( x ) , whereas the degree of P ( x ) + Q ( x ) is at most the maximum of degrees of P ( x ) and Q ( x ) .

Here is an example of the graph of a cubic polynomial function. – – – – – – – – – – x Figure . Suppose that f ( x ) and g ( x ) are polynomials where g ( x ) is not zero. The quotient f ( x ) g ( x ) is called a rational function, which is defined for all x ∈ R such that g ( x ) = .

In general, a rational function need not be a polynomial. . Polynomial Functions . .

Division Algorithm Given two polynomials f ( x ) and g ( x ) , where g ( x ) is not the zero polynomial, there exist two polynomials q ( x ) and r ( x ) such that f ( x ) = q ( x ) g ( x )+ r ( x ) where degree of r ( x ) < degree of g ( x ) . Here, q ( x ) is called the quotient polynomial, and r ( x ) is called the remainder polynomial. If r ( x ) is the zero polynomial, then q ( x ) , g ( x ) are factors of f ( x ) and f ( x ) = q ( x ) g ( x ) . These terminologies are similar to terminologies used in division done with integers.

If g ( x ) = x − a , then the remainder r ( x ) should have degree zero and hence r ( x ) is a constant. To determine the constant, write f ( x ) = ( x − a ) q ( x ) + c . Substituting x = a we get c = f ( a ) . Remainder Theorem If a polynomial f ( x ) is divided by x − a , then the remainder is f ( a ) .

Thus the remainder c = f ( a ) = if and only if x − a is a factor for f ( x ) . Definition . A real number a is said to be a zero of the polynomial f ( x ) if f ( a ) = . If x = a is a zero of f ( x ) , then x − a is a factor for f ( x ) .

In general, if we can express f ( x ) as f ( x ) = ( x − a ) k .g ( x ) where g ( a ) = , then the value of k , which depends on a , cannot exceed the degree of f ( x ) . The value k is called the multiplicity of the zero a . (i) A polynomial function of degree n can have at most n distinct real zeros. It is also possible that a polynomial function like P ( x ) = x + has no real zeros at all.

(ii) Suppose that P ( x ) is a polynomial function having rational coefficients. If a + b √ p where a, b ∈ Q , p a prime, is a zero of P ( x ) , then its conjugate a − b √ p is also a zero. Two important problems relating to polynomials are (i) Finding zeros of a given polynomial function; and hence factoring the polynomial into linear factors and (ii) Constructing polynomials with the given zeros and/or satisfying some additional conditions. To address the problem of finding zeros of a polynomial function, some well known algebraic identities are useful.

What is an identity? An equation is said to be an identity if that equation remains valid for all values in its domain. An equation is called conditional equation if it is true only for some (not all) of values in its domain. Let us recall the following identities.

. . Important Identities For all x, a, b ∈ R we have . ( x + a ) = ( x + a )