How do you use polynomial division to determine whether x=3 is a root of $x^{3} + x^{2} + 15 - x$ $x^3+x^2+15-x$ ?

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How do you use polynomial division to determine whether x=3 is a root of $x^{3} + x^{2} + 15 - x$ $x^3+x^2+15-x$ ?

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Answer 1 · 2015-11-24T17:51:39

Divide $(x^{3} + x^{2} - x + 15) \div (x - r)$ $(x^3 + x^2 -x + 15) -: (x-r)$ with long polynomial division.

If you can divide without a remainder, $x = r$ $x = r$ is a root.
If you have a remainder, it's not a root.

Explanation:

Well, just to check if $x = 3$ $x = 3$ is a root of your term, it would be easier to plug $x = 3$ $x = 3$ and see if the result is $0$ $0$ . :-)

But of course, it is also possible to determine the outcome with polynomial division.

To do so, you must divide $(x^{3} + x^{2} - x + 15)$ $(x^3 + x^2 -x + 15)$ by $(x - 3)$ $(x-3)$ .

I know that in some countries, a different notation for long division is being used but I will use the one that I'm familiar with and I hope that it will be no problem for you to rewrite it in your notation if necessary.

$\times (x^{3} + x^{2} \times - x \times + 15) : (x - 3) = x^{2} + 4 x + 11$ $color(white)(xx)(x^3 + x^2 color(white)(xx) - x color(white)(xx) + 15) : (x-3) = x^2 + 4x + 11$
$i - (x^{3} - 3 x^{2})$ $color(white)(i)-(x^3 - 3x^2)$
$\times x \frac{\times \times \times}{x}$ $color(white)(xxx)color(white)(xxxxxx)/color(white)(x)$
$\times \times \times x 4 x^{2} x - x$ $color(white)(xxxxxxx) 4 x^2 color(white)(x)- x$
$\times \times x - (4 x^{2} - 12 x)$ $color(white)(xxxxx) -(4 x^2 - 12x)$
$\times \times \times x \frac{\times \times \times \times}{x}$ $color(white)(xxxxxxx)color(white)(xxxxxxxx)/color(white)(x)$
$\times \times \times \times \times \times 11 x x + 15$ $color(white)(xxxxxxxxxxxx) 11x color(white)(x) + 15$
$\times \times \times \times \times - (11 x x - 33)$ $color(white)(xxxxxxxxxx) -(11x color(white)(x) - 33)$
$\times \times \times \times \times \times \frac{\times \times \times \times}{x}$ $color(white)(xxxxxxxxxxxx)color(white)(xxxxxxxx)/color(white)(x)$
$\times \times \times \times \times \times \times \times \times 48$ $color(white)(xxxxxxxxxxxxxxxxxx) 48$

At the end of the computation, there is the remainder $48$ $48$ .

As you have a remainder, it means that $x = 3$ $x = 3$ is not a root of $x^{3} + x^{2} - x + 15$ $x^3 + x^2 -x + 15$ .

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How do you use polynomial division to determine whether x=3 is a root of $x^{3} + x^{2} + 15 - x$ $x^3+x^2+15-x$ ?

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How do you use polynomial division to determine whether x=3 is a root of $x^{3} + x^{2} + 15 - x$ $x^3+x^2+15-x$ ?