Question

Using synthetic division, how to divide the following? (A) `(x^2+8x+11)÷ (x+3)` (B) `(x^3-4x^2-2x+5) ÷ (x-1)` (C) `(3x^4-4x+2x^3-1)÷(x+3)` (D) `(x^5-36) ÷ (x-2)`

Answer 1

`(B)(x^3-4x^2-2x+5)=(x-1)(x^2-3x-5)+0`

Explanation:

There are four questions in your one question.
Let us take `(B)(x^3-4x^2-2x+5)div(x-1)`
Here,

`p(x)=x^3-4x^2-2x+5and` divisor `x=1`

We take coefficients of `p(x)to1,-4,-2,5`
and set the problem as shown below.

`color(red)("Put zero {0} below the first number {1} and add : " 1+0=1`

. `1 |` `color(red)1color(white)(.....)-4color(white)(.....)-2color(white)(......)5`
`ulcolor(white)(.2)|` `ul(color(red)0color(white)(..................................)`
`color(white)(....)color(red)1`

`color(blue)("Now multiply this {1}with divisor"` `color(blue)((1)to1xx(1)=1` `color(blue)("and put below second number{-4} and add"`

`color(blue)(to-4+1=-3`

. `1 |` `1color(white)(.....)color(blue)(-4)color(white)(.....)-2color(white)(.......)5`
`ulcolor(white)()|` `ul(0color(white)(..........)color(blue)(1)color(white)(......................)`
`color(white)(....)1color(white)(.......)color(blue)(-3color(white)(.....20color(white)(.........)ul|0|`
Again repeat the process :

`i.e. color(brown)(-3xx(1)=-3 and -2+(-3)=-5`

.`1 |` `1color(white)(.........)-4color(white)(.......)color(brown)-2color(white)(.......)5`
`ulcolor(white)()|` `ul(0color(white)(...........)1color(white)(..........)color(brown)(-3)color(white)(.........10`
`color(white)(....)1color(white)(.........) -3color(white)(.........)color(brown)(-5)color(white)(.........ul|0|`

Again , `color(violet)(-5xx(1)=-5 and (5)+(-5)=0`

.`1 |` `1color(white)(.......)-4color(white)(.......)-2color(white)(........)color(violet)(5`
`ulcolor(white)()|` `ul(0color(white)(...........)1color(white)(.......)-3color(white)(......)color(violet)(-5`
`color(white)(....)1color(white)(.......)-3color(white)(.......)-5color(white)(.........)color(violet)(ul|0|`
We can see that , quotient polynomial :

`q(x)=x^2-3x-5 and"the Remainder"=0`

Hence ,

`(x^3-4x^2-2x+5)=(x-1)(x^2-3x-5)+0`

Note: For `(A),(C),and (D)" please see the second answer."`

Answer 2

`(A)(x^2+8x+11)=(x+3)(x+5)+(-4)`
`(C)(3x^4+2x^3-4x-1)`=`(x+3)(3x^3-7x^2+21x-67)+200`
`(D)(x^5-36)=(x-2)(x^4+2x^3+4x^2+8x+16)+(-4)`

Explanation:

`(A)(x^2+8x+11)div(x+3)`

We have , `p(x)=x^2+8x+11 and "divisor :"x=-3`

We take ,coefficients of `p(x) to1, 8, 11`

`-3 |` `1color(white)(.......)8color(white)(.......)11`
`ulcolor(white)(....)|` `ul(0color(white)( ...)-3color(white)(..)-15`
`color(white)(......)1color(white)(.......)5color(white)(.......)color(violet)(ul|-4|`
We can see that , quotient polynomial :

`q(x)=x+5 and"the Remainder"=-4`

Hence ,

`(x^2+8x+11)=(x+3)(x+5)+(-4)`
........................................................................................................

`(C)(3x^4+2x^3-4x-1)div(x+3)`

`p(x)=3x^4+2x^3color(red)(+0x^2)-4x-1to3,2,0,-4,-1`

`-3 |` `3color(white)(.......)2color(white)(.......)0color(white)(.......)-4color(white)(.......)-1`
`ulcolor(white)(....)|` `ul(0color(white)( ...)-9color(white)(.......)21color(white)(......)-63color(white)(.....)201`
`color(white)(......)3color(white)(....)-7color(white)(......)21color(white)(.......)-67color(white)(....)color(violet)(ul|200|`
Hence,
`(3x^4+2x^3-4x-1)` `=(x+3)(3x^3-7x^2+21x-67)+200`
................................................................................................................

`(D)(x^5-36)div(x-2)`

`p(x)=x^5+0x^4+0x^3+0x^2+0x-36to1,0,0,0,0,-36`

.`2 |` `1color(white)(.......)0color(white)(.......)0color(white)(.......)0color(white)(.......)0color(white)(...)-36`
`ulcolor(white)()|` `ul(0color(white)( .......)2color(white)(.......)4color(white)(.......)8color(white)(.....)16color(white)(......)32`
`color(white)(......)1color(white)(......)2color(white)(.......)4color(white)(.......)8color(white)(.......)16 color(white)(....)color(violet)(ul|-4|`
Hence,

`(x^5-36)=(x-2)(x^4+2x^3+4x^2+8x+16)+(-4)`