Remainder Theorem...2

[nycmathguy]

Section 2.3
Question 46

Can you do 46 for me? Show steps. Thanks.

 

[MathLover1]

46.
f(x)=x^3 -4x^2-10x+8 , k=-2

if zer is at k=-2, one factor of the given polynomial is (x-(-2))=(x+2)

and you can divide x^3 -4x^2-10x+8 by (x+2)

use long division


..........(x^2 -6x+2 ->Q(x)
(x+2)| x^3 -4x^2-10x+8
..........x^3+2x^2.........................subtract
.................-6x^2.........bring down next term
.................-6x^2-10x
.................-6x^2-12x.......................subtract
............................2x.......bring down next term
............................2x+8
............................2x+4
...................................4->reminder r

so, x^3 -4x^2-10x+8 = (x^2 -6x+2)(x + 2) +4

 

[nycmathguy]

46.
f(x)=x^3 -4x^2-10x+8 , k=-2

if zer is at k=-2, one factor of the given polynomial is (x-(-2))=(x+2)

and you can divide x^3 -4x^2-10x+8 by (x+2)

use long division


..........(x^2 -6x+2 ->Q(x)
(x+2)| x^3 -4x^2-10x+8
..........x^3+2x^2.........................subtract
.................-6x^2.........bring down next term
.................-6x^2-10x
.................-6x^2-12x.......................subtract
............................2x.......bring down next term
............................2x+8
............................2x+4
...................................4->reminder r

so, x^3 -4x^2-10x+8 = (x^2 -6x+2)(x + 2) +4

Why use long division when the instructions state to apply synthetic division?

 

[MathLover1]

in previous problem I did synthetic division step-by-step so you can learn how it is done

here is short version which you will do when you learn the steps


Write down the first coefficient without changes:

-2|1 -4 -10 8
| -2 12 -4
_|____________________
|1 -6 2 4

We have completed the table and have obtained the following resulting coefficients: 1,-6,2,4.
All the coefficients except the last one are the coefficients of the quotient, the last coefficient is the remainder.
Thus, the quotient is x^2-6x+2, and the remainder is 4.

 

[nycmathguy]

in previous problem I did synthetic division step-by-step so you can learn how it is done

here is short version which you will do when you learn the steps


Write down the first coefficient without changes:

-2|1 -4 -10 8
| -2 12 -4
_|____________________
|1 -6 2 4

We have completed the table and have obtained the following resulting coefficients: 1,-6,2,4.
All the coefficients except the last one are the coefficients of the quotient, the last coefficient is the remainder.
Thus, the quotient is x^2-6x+2, and the remainder is 4.

Hello. I know the short version of synthetic division. I learned it years ago. Synthetic division only works when the divisor is of the form (x - a), where a is an integer.