Question

How do you divide `(x^2-6x+4)div(x+1)` using long division?

Answer 1

`x-7+frac{11}[x+1}`

Explanation:

`color(white)(aaaaaaaaaaaaa)color(pink)xcolor(blue)(-7)`
`color(white)(aaaaa)`------------------------
`color(green)(x+1)|~color(white)(aa)x^2-6x+4`
`color(white)(aaaa)-(x^2+x)`
`color(white)(aaaaaaa)`-------------
`color(white)(aaaaaaaaa)-7x+4`
`color(white)(aaaaaa)-(-7x-7)`
`color(white)(aaaaaaaaaa)`--------------
`color(white)(aaaaaaaaaaaaaaaa)color(red)(11)`

To find the first term of the quotient, divide the first term of the dividend (`x^2`) by the first term of the divisor (`x`). `x^2/x=color(pink)x` Write the `color(pink)x` over the `-6x` term of the dividend.

Multiply the `color(pink)x` by the divisor `color(green)( x+1 )`. Write the product `x^2+x` under the dividend, then subtract it from the dividend.
Write the difference `-7x` under the line as shown and "pull down" the last term `4`.

Divide the `-7x` by the first term of the divisor `x`. Write the quotient `color(blue)(-7)` over the 4 term of the dividend.

Multiply the `color(blue)(-7)` by the divisor `color(green)(x+1)`. Write the product `-7x-7` under the `-7x+4` and subtract. The difference `color(red)(11)` is the remainder.

With a remainder of `color(red)(11)` the answer is
`x-7+frac{color(red)(11)}{x+1}`

Answer 2

`(x^2-6x+4)div(x+1) = (x-7) " rem "11`

Or `(x^2-6x+4)div(x+1) = (x-7) +11/(x+1)`

Explanation:

The algebraic long division follows the same method as arithmetic long division...

`("dividend")/("divisor") = "quotient"`

1. Write the dividend in the 'box' making sure that the indices are in descending powers of x.

2. Divide the first term in divisor into the term in the dividend with the highest index. Write the answer at the top,

3. Multiply by BOTH terms of the divisor at the side

4. Subtract

5. Bring down the next term

REpeat steps 2 to 5

`color(white)(xxxxxxxxxxxx)color(red)(x)color(blue)(-7)`
`color(white)(xxx)x+1 |bar( x^2 -6x +4)" "larr x^2divx = color(red)(x)`
`color(white)(xxxxxxxx)ul(color(red)(x^2+1x))" "darr" "larr` subtract (change signs)
`color(white)(xxxxxxxxxx) -7x+4""larr` bring down the 4, `-7x div x = color(blue)(-7)`
`color(white)(xxxxxxxxxx.)ul(color(blue)(-7x-7))" "larr` subtract (change signs)
`color(white)(xxxxxxxxxxxxxxx)11 " "larr`remainder

`(x^2-6x+4)div(x+1) = (x-7) " rem "11`

This can also be written as

`(x^2-6x+4)div(x+1) = (x-7) +11/(x+1)`