Question

How do you divide `(x^3-11x^2+22x+40)div(x-5)` using synthetic division?

Answer 1

`x^2-6x-8`

Explanation:

The coefficients of the first polynomial become the dividend.
The "zero" of the second polynomial becomes the divisor. In other words, solve `x-5=0` and use the result (5) as the divisor.

`5__|1color(white)(aa)-11color(white)(aaaa)22color(white)(aaa)40`
`color(white)(aaAaaaaa)5color(white)(a)-30color(white)(a)-40`
`color(white)(aa)`-----------------------------------
`color(white)(aa)color(red)1color(white)(aaa)color(red)(-6)color(white)(aa)color(red)(-8)color(white)(aaaa)color(red)0`

To do synthetic division, "pull down" the first coefficient (`1`) below the line. Multiply the divisor by the first coefficient. `5*1=5`

Write the product (`5`) under the next coefficient and add. (`-11+5=-6`) Write the sum under the line.

Multiply the divisor by the new number under the line.
(`5*-6=-30`)

Write the product (-30) under the next coefficient and add `22+ -30=-8`

Continue as above. The remainder is zero.

Then, use the numbers you've written under the line as the coefficients of the quotient. Start with a variable of degree one less than the degree of the dividend.

The quotient is:
`color(red)1x^2color(red)(-6)xcolor(red)(-8)+ color(red)0/(x^3-11x^2+22x+40)`

The last term represents the remainder and is equal to zero.

Thus, the quotient is:
`x^2-6x-8`