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# Polynomial Function | Remainder Theorem, Factor Theorem, and Synthetic Division

July 21, 2022 written by Azhar Ejaz

A polynomial function in x is an expression of the form

anx n+an-1 xn-1+….+a1nx+a0,     a≠0 –> (i)

Where n is a non-negative integer and the coefficients an,an-1,…,a1 and a0 are real numbers.

It can be considered as a polynomial function of the highest power of x in a polynomial in x is called the degree of the polynomial. So the expression (i), is a polynomial of degree n.

For example:

The polynomials x2-3x+2, 3x3+2x2-5x+4 are degree 2 and 3 respectively.

Dividend =(divisor)(quotient)+remainder

## Remainder Theorem:

If a polynomial f(x) of degree n≥1, n is a non-negative integer divided by (x-a) til no x-term exists in the remainder, then f(a) is the remainder.

Proof: suppose we divide a polynomial f(x) by x-a.Then there exists a unique quotient q(x) and a unique remainder R such that f(x)=(x-a)q(x)+R–>(i)

Substituting x=a in equation (i), we get

f(a)= (a-a)q(a)+R

f(a) =0+R

f(a)=R

Hence reminder =f(a)

A reminder obtained when f(x) is divided by (x-a) is the same as the value of the polynomial f(x) at x=0

For example: Find the remainder when the polynomial x3+4x2-2x+5 is divided by x-1.

Solution: let f(x)= x3+4x2-2x+5 and

x-a=x-1

we have

a=1

Remainder =f(1)                   (by remainder theorem )

Remainder =(1)3+4(1)2-2(1)+5

Remainder =1+4-2+5

Remainder =8

For example: find the numerical value of K if the polynomial x3+kx2-7x+6 has a remainder of -4 when divided by x+2.

Solution: let f(x)= x3+kx2-7x+6  and x-a=x+2, we have a=-2

Remainder =f(-2)                 (by remainder theorem )

Remainder =(-2)3+k(-2)2-7(-2)+6

Remainder = -8+4k+14+6

Remainder =4k+12

Given that remainder =-4

4k+12=-4

4k=-4-12

4k=-16

K=-16/4

K=-4

## Factor Theorem:

The polynomial( x-a) is a factor of the polynomial f(x) if and only if f(a)=0 i.e (x-a) is a factor of f(x) if and only if x=a is a root of the polynomial equation f(x)=0.

Proof: suppose g(x) is the quotient and R is the remainder when a polynomial f(x) is divided by(x-a) then by the remainder theorem

f(x)=(x-a)g(x)+R

Since f(a)=0

R=0

f(x)=(x-a)g(x)+0

f(x)= (x-a)g(x)

(x-a) is a factor of f(x).

Conversely, if (x-a) is a factor of f(x), then

R=f(a)=0

Hence proved

To determine if a given linear polynomial (x-a) is a factor of f(x), all we need to check is whether f(a)=0

For example: show that (x-2) is a factor of x4-13x2+36

Solution: let f(x)= x4-13x2+36 and x-a=x-2 we have x=2

Remainder =f(2)                     (by remainder theorem )

Remainder=(2)4-13(2)2+36

Remainder =16-52+36

Remainder =52-52

Remainder =0

(x-2) is a factor of x4-13x2+36

## Synthetic Division:

There is a shortcut method for the long division of a polynomial f(x) by a polynomial of the form(x-a). This process of division is called synthetic division.

### Outline of the Method:

• Write down the coefficients of the dividend f(x) from left to right in decreasing order of powers of x .insert 0 for any missing terms.
• To the left of the first line, write an of the divisor (x-a).
• Use the following patterns to write the second and third lines :

Vertical pattern (↓)

Diagonal pattern (↗)

Multiply by a

For example:

Use synthetic division to find the quotient and the remainder when the polynomial x4-10x2-2x+4 is divided by x+3.

Solution: let f(x)= x4-10x2-2x+4

f(x)= x4 +0x3-10x2-2x+4

And x-a=x-(-3)

x=-3

Dividend x4-10x2-2x+4 Quotient =x3-3×2-x+1 Remainder =1

### What is meant by a Polynomial function?

A polynomial function is a mathematical function that can be expressed in the form of a polynomial. It has a general form of P(x) = anxn + an – 1xn – 1 + … + a2x2 + a1x + ao, where exponent on x is any positive integer and ai’s are real numbers; i = 0, 1, 2, …, n

### What is the Remainder Theorem Formula?

The general formula for the remainder theorem is as follows: when p(x) is divided by (ax + b), the remainder = p(-b/a).

### What is the Use of the Factor Theorem?

The Factor Theorem is an important tool in mathematics that allows us to find the factors of a given polynomial equation. In other words, if we have a polynomial equation f(x), and we want to find out what factors into it, we can set f(k) = 0 and then (x – k) will be a factor of f(x).

### What is Factor Theorem?

The factor theorem states that if f(x) is a polynomial of degree n greater than or equal to 1, and ‘a’ is any real number, then (x – a) is a factor of f(x) if f(a) = 0. It is mainly used to factor the polynomials and to find the n roots of the polynomials

### Where do we Use the Factor Theorem in Real Life?

Factoring can be useful in many everyday situations, such as exchanging money, dividing quantities, understanding time, and comparing prices.

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