A polynomial is a function defined for all real *x* involving positive powers of *x* in the form:

...assuming that *P(x)* is a continuous and differentiable function.

The above polynomial expression has **degree** *n* where . While the *p's* are called **coefficients**. is called the **leading term** and the **leading coefficient**, while is called the constant term.

If , *P(x)* is called a **monic** polynomial.

While if then *P(x)* is the zero polynomial.

### Examples

### Show Answer

The real values of *x* that satisfy the equation are called the real **roots** of the equation or the real **zeroes** of the polynomial.

### Examples

1. Show that the polynomial has no real zeroes.

Use discriminant:

So the polynomial has no real zeroes.

2. For the polynomial

a) Evaluate *a* if the polynomial has real zeroes.

b) Find the degree of the derivative

### Show Answer

a) For monic polynomial, *a* = 1

b)

has degree 4 (highest power).

## Division of Polynomials

A polynomial *P(x)* can be written as where *P(x)* is the **dividend**, A*(x)* is the **divisor**, Q*(x)* is the **quotient**, and *R(x)* is the **remainder**.

The degree of remainder R(x) is always less than the degree of the divisor *A(x)*.

For example, divide by .

This means that the .

**Synthetic Division of Polynomials** is a shortcut method for dividing a polynomial by a linear factor. It is a condensed form of the long division process whenever a polynomial is divided by a first degree polynomial *x - a*.

For example, let us divide , the leading coefficient is 2. Since the term is missing, which means its coefficient is zero.

Applying synthetic division as below:

Multiply 3 by the divisor 2 giving us 6. We add 0 and get 6. The result is shown below on the right. An continuing the process, we get:

The coefficient of the last row is 2, 6, 19, 52 and a remainder of 158.

Therefore, the result will be: .

Another example:

## Remainder Theorem

If a polynomial *P(x)* is divided by *x - a*, then the remainder is *P(a)*.

### Examples

1. Find the remainder when is divided by *x - 2*.

### Show Answer

By remainder theorem,

So the remainder is 51.

2. Evaluate *m* if the remainder is 4 when dividing by *x + 3*.

### Show Answer

The remainder when *P(x)* is divided by *x + 3* is *P(-3)*. So,

From these two examples then we can say:

For polynomial *P(x)*, if *P(a)* = 0 then *x - a* is a factor of the polynomial.

And polynomial *P(x)*, if *x - a* i a factor of the polynomial, then *P(a)* = 0.

## More Properties of Polynomial

If polynomial *P(x)* has degree *n* and *n* distinct zeroes , then .

A polynomial of degree *n* cannot have more than *n* distinct real zeroes.

If two polynomials of degree *n* are equal for more than *n* distinct values of *x*, then the coefficients of like powers of *x* are equal. That is:

If *x - a* is a factor of polynomial *P(x)*, then *a* is a factor of the constant term of the polynomial.

### Examples

1. Find all factors of .

### Show Answer

Try factors of -12 (i.e.

Therefore *x* - 2 is a factor of *f(x)*. We can use this to find other factors:

Therefore,

2. Write in the form .