How do you use the remainder theorem and Synthetic Division to find the remainders in the following division problems $x^3 - 2x^2 + 5x - 6$ divided by x - 3?

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Answer 1 · 2016-12-15T14:30:09

The quotient is $=(x^2+x+8)$ and the remainder is $=18$

When we divide a polynomial $f(x)$ by $(x-c)$ , we get

$f(x)=(x-c)q(x)+r(x)$

Let $x=c$ , then $f(c)=r$

Here, $f(x)=x^3-2x^2+5x-6$ and $c=3$

Therefore,

$f(3)=3^3-2*3^2+5*3-6=27-18+15-6=18$

The remainder is $=18$

Let's do the long division

$color(white)(aaaa)$ $x^3-2x^2+5x-6$ $color(white)(aaaa)$ $∣$ $x-3$

$color(white)(aaaa)$ $x^3-3x^2$ $color(white)(aaaaaaaaaaaa)$ $∣$ $x^2+x+8$

$color(white)(aaaaa)$ $0+x^2+5x$

$color(white)(aaaaaaaa)$ $x^2-3x$

$color(white)(aaaaaaaaa)$ $0+8x-6$

$color(white)(aaaaaaaaaaa)$ $+8x-24$

$color(white)(aaaaaaaaaaaaaa)$ $0+18$

The quotient is $=(x^2+x+8)$ and the remainder is $=18$

How do you use the remainder theorem and Synthetic Division to find the remainders in the following division problems x^3 - 2x^2 + 5x - 6x^3 - 2x^2 + 5x - 6 divided by x - 3?