Question

How do you use synthetic division and Remainder Theorem to find P(a) if `P(x) = x^3 - 8x^2 + 5x - 7` and a = 1?

Answer 1

`P(1)=-9`

Explanation:

Set up the synthetic division problem with the coefficients of the polynomial as the dividend and the `color(red)1` as the divisor.

`color(red)1|color(white)(aa)1color(white)(aaa)-8color(white)(aaaa)5color(white)(aa)-7`
`color(white)(aaaa)darr`
`color(white)(aaa^2a)color(blue)1color(white)(aaaaaaaaaaaaaaaaaaa)`Pull down the `color(blue)1`

`color(red)1|color(white)(aa)1color(white)(aaa)-8color(white)(aaaa)5color(white)(aa)-7`
`color(white)(aaaa)darrcolor(white)(aaaaa)color(limegreen)1color(white)(aaaaaaaaaaaaa)`Multiply `color(red)1 *color(blue)1` and write the product
`color(white)(aaa^2a)color(blue)1color(white)(aa^(2)a)color(blue)(-7)color(white)(aaaaaaaaaaaaacolor(limegreen)1` under the 8. Add `-8+color(limegreen)1=color(blue)(-7)`

`color(red)1|color(white)(aa)1color(white)(aaa)-8color(white)(aaaa)5color(white)(aa)-7`
`color(white)(aaaa)darrcolor(white)(aaaaa)color(limegreen)1color(white)(a^(22))color(limegreen)(-7)color(white)(aaaaAa)`Multiply `color(red)1*color(blue)(-7)` and put the product
`color(white)(aaa^2a)color(blue)1color(white)(aaaa)color(blue)(-7)color(white)(a^2a)color(blue)(-2)color(white)(aaaaaa)color(limegreen)(-7)` under the `5`. Add `5+color(limegreen)(-7)=color(blue)(-2)`

`color(red)1|color(white)(aa)1color(white)(aaa)-8color(white)(aaaa)5color(white)(aa)-7`
`color(white)(aaaa)darrcolor(white)(aaaaa)color(limegreen)1color(white)(aa)color(limegreen)(-7)color(white)(aaa)color(limegreen)(-2)color(white)(aa)`Repeat multiplying and adding
`color(white)(aaa^2a)color(blue)1color(white)(aa^(2)a)color(blue)(-7)color(white)(a^11)color(blue)(-2)color(white)(aaa)color(magenta)(-9)`

Note that the last number or remainder is `color(magenta)(-9)`.

According to the remainder theorem, `P(1)=color(magenta)(-9)`