How do you divide $\frac{x^{3} + x + 3}{x - 5}$ $(x^3+x+3)/(x-5)$ ?

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How do you divide $\frac{x^{3} + x + 3}{x - 5}$ $(x^3+x+3)/(x-5)$ ?

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Answer 1 · 2017-10-28T14:09:55

The remainder is $133$ $color(red)(133)$ and the quotient is $= x^{2} + 5 x + 26$ $=x^2+5x+26$

Explanation:

Let's perform a synthetic division

$a a$ $color(white)(aa)$ $5$ $5$ $a a a a a$ $color(white)(aaaaa)$ $∣$ $|$ $a a a$ $color(white)(aaa)$ $1$ $1$ $a a a a a$ $color(white)(aaaaa)$ $0$ $0$ $a a a a a a$ $color(white)(aaaaaa)$ $1$ $1$ $a a a a a a a a a a$ $color(white)(aaaaaaaaaa)$ $3$ $3$
$a a a a a a a a a a a a$ $color(white)(aaaaaaaaaaaa)$ $- - - - - - - - - - - -$ $------------$

$a a a a$ $color(white)(aaaa)$ $a a a a$ $color(white)(aaaa)$ $∣$ $|$ $a a a a$ $color(white)(aaaa)$ $a a a a a$ $color(white)(aaaaa)$ $5$ $5$ $a a a a a a$ $color(white)(aaaaaa)$ $25$ $25$ $a a a a a a a a$ $color(white)(aaaaaaaa)$ $130$ $130$
$a a a a a a a a a a a a$ $color(white)(aaaaaaaaaaaa)$ $- - - - - - - - - - - -$ $------------$

$a a a a$ $color(white)(aaaa)$ $a a a a$ $color(white)(aaaa)$ $∣$ $|$ $a a a$ $color(white)(aaa)$ $1$ $1$ $a a a a a$ $color(white)(aaaaa)$ $5$ $5$ $a a a a a a$ $color(white)(aaaaaa)$ $26$ $26$ $a a a a a a a a$ $color(white)(aaaaaaaa)$ $133$ $color(red)(133)$

The remainder is $133$ $color(red)(133)$ and the quotient is $= x^{2} + 5 x + 26$ $=x^2+5x+26$

$\frac{x^{3} + x + 3}{x - 5} = x^{2} + 5 x + 26 + \frac{133}{x - 5}$ $(x^3+x+3)/(x-5)=x^2+5x+26+133/(x-5)$

Answer 2 · 2017-10-28T14:55:28

$x^{2} + 5 x + 26 + \frac{133}{x - 5}$ $x^2+5x+26+133/(x-5)$

Explanation:

$one way is to use the divisor as a factor in the numerator$ $"one way is to use the divisor as a factor in the numerator"$

$consider the numerator$ $"consider the numerator"$

$x^{2} (x - 5) + 5 x^{2} + x + 3$ $color(red)(x^2)(x-5)color(magenta)(+5x^2)+x+3$

$= x^{2} (x - 5) + 5 x (x - 5) + 25 x + x + 3$ $=color(red)(x^2)(x-5)color(red)(+5x)(x-5)color(magenta)(+25x)+x+3$

$= x^{2} (x - 5) + 5 x (x - 5) + 26 (x - 5) + 130 + 3$ $=color(red)(x^2)(x-5)color(red)(+5x)(x-5)color(red)(+26)(x-5)color(magenta)(+130)+3$

$= x^{2} (x - 5) + 5 x (x - 5) + 26 (x - 5) + 133$ $=color(red)(x^2)(x-5)color(red)(+5x)(x-5)color(red)(+26)(x-5)+133$

$quotient = x^{2} + 5 x + 26, remainder = 133$ $"quotient "=color(red)(x^2+5x+26)," remainder" =133$

$\Rightarrow \frac{x^{3} + x + 3}{x - 5} = x^{2} + 5 x + 26 + \frac{133}{x - 5}$ $rArr(x^3+x+3)/(x-5)=x^2+5x+26+133/(x-5)$

Answer 3 · 2017-10-28T16:24:47

Quotient $= x^{2} + 5 x + 26$ $=x^2+5x+26$ and remainder $\frac{133}{x - 5}$ $133/(x-5)$

Explanation:

$color(white)(..........)color(white)(.)x^2+5x+26$
$x-5|overline(x^3+0+x+3)$
$color(white)(............)ul(x^3-5x^2)$
$color(white)(......................)5x^2+x$
$color(white)(......................)ul(5x^2-25x)$
$color(white)(..................................)26x+3$
$color(white)(..................................)ul(26x-130)$
$color(white)(................................................)133$

$(x^3+x+3) / (x-5) = x^2+5x+26+133/(x-5)$

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