How do you use synthetic division to show that x=1/2 is a zero of $2x^3-15x^2+27x-10=0$ ?

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Answer 1 · 2018-07-12T16:11:35

Please see the explanation below

Divide by $1/2$

$color(white)(aaaa)$ $1/2$ $|$ $color(white)(aaaa)$ $2$ $color(white)(aaaa)$ $-15$ $color(white)(aaaaaa)$ $27$ $color(white)(aaaaaa)$ $-10$

$color(white)(aaaaaa)$ $|$ $color(white)(aaaa)$ $color(white)(aaaaaaa)$ $1$ $color(white)(aaaaa)$ $-7$ $color(white)(aaaaaaaa)$ $10$

$color(white)(aaaaaaaaa)$ _________________________________________________________##

$color(white)(aaaaaa)$ $|$ $color(white)(aaaa)$ $2$ $color(white)(aaaa)$ $-14$ $color(white)(aaaaaa)$ $20$ $color(white)(aaaaaaaa)$ $color(red)(0)$

The remainder is $=(0)$ and the quotient is $=(2x^2-14x+20)$

As the remainder $=0$ , $x=1/2$ is a root of $2x^3-15x^2+27x-10$

$2x^3-15x^2+27x-10=(2x^2-14x+20)(x-1/2)$

How do you use synthetic division to show that x=1/2 is a zero of 2x^3-15x^2+27x-10=02x^3-15x^2+27x-10=0?