Remainder theorem calculator

SolutionHelp

Solution

Solution provided by AtoZmath.com

1. `x^2-10x+25` and `x-5`

2. `x^3+4x^2-4x-16` and `x-2`

3. `x^3+6x^2+12x+10` and `x+2`

Example

1. Find remainder theorem of `x^2-10x+25` and `x-5`

Solution:
`(x^2-10x+25)/(x-5)` using remainder theorem

Remainder theorem : If an expression `f(x)` is divided by `(x-a)` then the remainder is `f(a)`.

To determine root divisor, we have to solve divisor equation `x-5=0`

`:.x=5`

So `f(x)=x^2-10x+25`

we have to find `f(5)`

`f(5)= 5^2-10 xx 5+25`

`=25-50+25`

`=0`

`:.` The remainder is `0`.

Method-2: solution using synthetic division method

To determine root divisor, we have to solve divisor equation `x-5=0`

`:.` our root becomes `x=5`

Write coefficients of the dividend `x^2-10x+25` to the right and our root `5` to the left

`5`	`1`	`-10`	`25`
	``	``	``

Step-1 : Write down the first coefficient `1`

`5`	`1`	`-10`	`25`
	``	``	``
	`1`

Step-2 : Multiply our root `5` by our last result `1` to get `5` [ `5` × `1`=`5` ]

`5`	`1`	`-10`	`25`
	``	`5`	``
	`1`

Step-3 : Add new result `5` to the next coefficient of the dividend `-10`, and write down the sum `-5`, [ `(-10)` + `5`=`-5` ]

`5`	`1`	`-10`	`25`
	``	`5`	``
	`1`	`-5`

Step-4 : Multiply our root `5` by our last result `-5` to get `-25` [ `5` × `(-5)`=`-25` ]

`5`	`1`	`-10`	`25`
	``	`5`	`-25`
	`1`	`-5`

Step-5 : Add new result `-25` to the next coefficient of the dividend `25`, and write down the sum `0`, [ `25` + `(-25)`=`0` ]

`5`	`1`	`-10`	`25`
	``	`5`	`-25`
	`1`	`-5`	`0`

We have completed the table and have obtained the following coefficients
`1,-5,0`

All coefficients, except last one, are coefficients of quotient, last coefficient is remainder.
Thus quotient is `x-5` and remainder is `0`