If a polynomial f(x) is divided by x−a, then the remainder is f(a). Thus the remainder c = f(a) = 0 if and only if x − a is a factor for f(x).

**Polynomial Functions**

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*) = 0 if and only if *x* *−* *a* is a factor for *f*(*x*).

In general, if we can
express *f*(*x*) as *f*(*x*) = (*x*
*−* *a*)* ^{k}.g*(

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

Now let us focus on constructing
polynomials with the given information using the method of undetermined
coefficients. That is, we shall determine coefficients of the required
polynomial using the given conditions. The main idea here is that two
polynomials are equal if and only if the coefficients of same powers of the
variables in the two polynomials are equal.

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