How do you divide $3 x^{4} + 5 x^{3} - x^{2} + x - 2$ $3x^4+5x^3-x^2+x-2$ by $x - 2$ $x-2$ ?

Question

How do you divide $3 x^{4} + 5 x^{3} - x^{2} + x - 2$ $3x^4+5x^3-x^2+x-2$ by $x - 2$ $x-2$ ?

2 Answers

Impact of this question

2123 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2015-09-11T02:09:04

See in the explanation

Explanation:

We have that

$3 x^{4} + 5 x^{3} - x^{2} + x - 2 = (3 x^{3} + 11 x^{2} + 21 x + 43) \cdot (x - 2) + 84$ $3x^4+5x^3-x^2+x-2=(3x^3+11x^2+21x+43)*(x-2)+84$

Answer 2 · 2015-09-11T18:02:33

See the explanation section.

Explanation:

$\frac{3 x^{4} + 5 x^{3} - x^{2} + x - 2}{x - 2}$ $(3x^4+5x^3-x^2+x-2)/(x-2)$

There are various ways of writing the details Here's one way.

$" " " "$ $- - - - - - - -$ $--------$
$x - 2)$ $x-2 )$ $3 x^{4}$ $3x^4$ $+ 5 x^{3}$ $+5x^3$ $- x^{2}$ $-x^2$ $+ x$ $+x$ $- 2$ $-2$

What do we need to multiply the first term on the divisor ( $x$ $x$ ) by to get the first term of the dividend ( $3 x^{4}$ $3x^4$ )? Clearly, we need to multiply by $3 x^{3}$ $3x^3$

$3 x^{3}$ $" " " " " " "3x^3$
$" " " "$ $- - - - - - - -$ $--------$
$x - 2)$ $x-2 )$ $3 x^{4}$ $3x^4$ $+ 5 x^{3}$ $+5x^3$ $- x^{2}$ $-x^2$ $+ x$ $+x$ $- 2$ $-2$

Now multiply $3 x^{3}$ $3x^3$ times the divisor, $x - 2$ $x-2$ , to get $3 x^{4} - 6 x^{3}$ $3x^4-6x^3$ and write that under the dividend.

$3 x^{3}$ $" " " " " " "3x^3$
$" " " "$ $- - - - - - - -$ $--------$
$x - 2)$ $x-2 )$ $3 x^{4}$ $3x^4$ $+ 5 x^{3}$ $+5x^3$ $- x^{2}$ $-x^2$ $+ x$ $+x$ $- 2$ $-2$
$" "" "" "$ $3 x^{4}$ $3x^4$ $- 6 x^{3}$ $-6x^3$
$" " " "$ $- - - - -$ $-----$

Now we need to subtract $3 x^{4} - 6 x^{3}$ $3x^4-6x^3$ from the dividend. (You may find it simpler to change the signs and add.)

$3 x^{3}$ $" " " " " " "" "3x^3$
$" " " "$ $- - - - - - - -$ $--------$
$x - 2)$ $x-2 )" "$ $3 x^{4}$ $3x^4$ $+ 5 x^{3}$ $+5x^3$ $- x^{2}$ $-x^2$ $+ x$ $+x$ $- 2$ $-2$
$" " " "$ $- 3 x^{4} + 6 x^{3}$ $color(red)(-)3x^4color(red)(+)6x^3$
$" "" "" "$ $- - - - -$ $-----$
$" "" "" "" "" "$ $11 x^{3}$ $11x^3$ $- x^{2}$ $-x^2$ $+ x$ $+x$ $- 2$ $-2$

Now, what do we need to multiply $x$ $x$ (the first term of the divisor) by to get $11 x^{3}$ $11x^3$ (the first term of the last line)? We need to multiply by $11 x^{2}$ $11x^2$
So write $11 x^{2}$ $11x^2$ on the top line, then multiply $11 x^{2}$ $11x^2$ times the divisor $x - 2$ $x-2$ , to get $11 x^{3} - 22 x^{2}$ $11x^3-22x^2$ and write it underneath.

$3 x^{3}$ $" " " " " " "" "3x^3$ $+ 11 x^{2}$ $+11x^2$
$" " " "$ $- - - - - - - -$ $--------$
$x - 2)$ $x-2 )" "$ $3 x^{4}$ $3x^4$ $+ 5 x^{3}$ $+5x^3$ $- x^{2}$ $-x^2$ $+ x$ $+x$ $- 2$ $-2$
$" " " "$ $- 3 x^{4} + 6 x^{3}$ $color(red)(-)3x^4color(red)(+)6x^3$
$" "" "" "$ $- - - - -$ $-----$
$" "" "" "" "" "$ $11 x^{3}$ $11x^3$ $- x^{2}$ $-x^2$ $+ x$ $+x$ $- 2$ $-2$
$" "" "" "" "" "$ $11 x^{3}$ $11x^3$ $- 22 x^{2}$ $-22x^2$
$" " " "" "" "$ $- - - - - -$ $------$

Now subtract (change the signs and add), to get:

$3 x^{3}$ $" " " " " " "" "3x^3$ $+ 11 x^{2}$ $+11x^2$
$" " " "$ $- - - - - - - -$ $--------$
$x - 2)$ $x-2 )" "$ $3 x^{4}$ $3x^4$ $+ 5 x^{3}$ $+5x^3$ $- x^{2}$ $-x^2$ $" "$ $+ x$ $+x$ $- 2$ $-2$
$" " " "$ $- 3 x^{4} + 6 x^{3}$ $color(red)(-)3x^4color(red)(+)6x^3$
$" "" "" "$ $- - - - -$ $-----$
$" "" "" "" "" "$ $11 x^{3}$ $11x^3$ $- x^{2}$ $-x^2$ $" "$ $+ x$ $+x$ $- 2$ $-2$
$" "" "" "" "$ $- 11 x^{3}$ $color(red)(-)11x^3$ $+ 22 x^{2}$ $color(red)(+)22x^2$
$" " " "" "" "$ $- - - - - -$ $------$
$" "" "" "" "" "" "" "" "$ $21 x^{2}$ $21x^2$ #+x-2#

Repeat to get $21 x$ $21x$ , so we put the $9$ $9$ on top multiply, subtract (change signs and add) to get:

$3 x^{3}$ $" " " " " " "" "3x^3$ $+ 11 x^{2}$ $+11x^2$ $+ 21 x$ $+21x$
$" " " "$ $- - - - - - - -$ $--------$
$x - 2)$ $x-2 )" "$ $3 x^{4}$ $3x^4$ $+ 5 x^{3}$ $+5x^3$ $- x^{2}$ $-x^2$ $" "$ $+ x$ $+x$ $- 2$ $-2$
$" " " "$ $- 3 x^{4} + 6 x^{3}$ $color(red)(-)3x^4color(red)(+)6x^3$
$" "" "" "$ $- - - - -$ $-----$
$" "" "" "" "" "$ $11 x^{3}$ $11x^3$ $- x^{2}$ $-x^2$ $" "$ $+ x$ $+x$ $- 2$ $-2$
$" "" "" "" "$ $- 11 x^{3}$ $color(red)(-)11x^3$ $+ 22 x^{2}$ $color(red)(+)22x^2$
$" " " "" "" "$ $- - - - - -$ $------$
$" "" "" "" "" "" "" "" "$ $21 x^{2}$ $21x^2$ #+x" "# $- 2$ $-2$
$" "" "" "" "" "" "" "$ $- 21 x^{2}$ $color(red)(-)21x^2$ $+ 42 x$ $color(red)(+)42x$
$" " " "" "" "" "" "$ $- - - - - - - -$ $--------$
$" "" "" "" "" "" "" "" "" "" "" "$ $43 x$ $43x$ $- 2$ $-2$
We'll be done when the last line is $0$ $0$ or has degree less than the degree of the divisor. Which has not happened yet, but we're close.

$3 x^{3}$ $" " " " " " "" "3x^3$ $+ 11 x^{2}$ $+11x^2$ $+ 21 x$ $+21x$ $+ 43$ $+43$
$" " " "$ $- - - - - - - -$ $--------$
$x - 2)$ $x-2 )" "$ $3 x^{4}$ $3x^4$ $+ 5 x^{3}$ $+5x^3$ $- x^{2}$ $-x^2$ $" "$ $+ x$ $+x$ $- 2$ $-2$
$" " " "$ $- 3 x^{4} + 6 x^{3}$ $color(red)(-)3x^4color(red)(+)6x^3$
$" "" "" "$ $- - - - -$ $-----$
$" "" "" "" "" "$ $11 x^{3}$ $11x^3$ $- x^{2}$ $-x^2$ $" "$ $+ x$ $+x$ $- 2$ $-2$
$" "" "" "" "$ $- 11 x^{3}$ $color(red)(-)11x^3$ $+ 22 x^{2}$ $color(red)(+)22x^2$
$" " " "" "" "$ $- - - - - -$ $------$
$" "" "" "" "" "" "" "" "$ $21 x^{2}$ $21x^2$ #+x" "# $- 2$ $-2$
$" "" "" "" "" "" "" "$ $- 21 x^{2}$ $color(red)(-)21x^2$ $+ 42 x$ $color(red)(+)42x$
$" " " "" "" "" "" "$ $- - - - - - - -$ $--------$
$" "" "" "" "" "" "" "" "" "" "" "$ $43 x$ $43x$ $- 2$ $-2$
$" "" "" "" "" "" "" "" "" "" "$ $- 43 x$ $color(red)(-)43x$ $+ 86$ $color(red)(+)86$
$" " " "" "" "" "" "$ $- - - - - - - -$ $--------$
$" "" "" "" "" "" "" "" "" "" "" "" "" "" "$ $84$ $84$

Now the last line has degree less than $1$ $1$ , so we are finished.

The quotient is: $3 x^{3} + 11 x^{2} + 21 x + 43$ $3x^3+11x^2+21x+43$ and the remainder is $84$ $84$

We can write:

$\frac{3 x^{4} + 5 x^{3} - x^{2} + x - 2}{x - 2} = 3 x^{3} + 11 x^{2} + 21 x + 43 + \frac{84}{x - 2}$ $(3x^4+5x^3-x^2+x-2)/(x-2)= 3x^3+11x^2+21x+43 + 84/(x-2)$

IMPORTANT to understanding what we have done:
If we get a common denominator on the right and simplify we will get exactly the left side.

Science

Math

Humanities

... and beyond

How do you divide $3 x^{4} + 5 x^{3} - x^{2} + x - 2$ $3x^4+5x^3-x^2+x-2$ by $x - 2$ $x-2$ ?

2 Answers

Explanation:

Explanation:

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

How do you divide 3x4+5x3−x2+x−23x^4+5x^3-x^2+x-2 by x−2x-2?

2 Answers

Explanation:

Explanation:

Related questions

Impact of this question

How do you divide $3 x^{4} + 5 x^{3} - x^{2} + x - 2$ $3x^4+5x^3-x^2+x-2$ by $x - 2$ $x-2$ ?