Divide Using Long Polynomial Division (x^5+243)/(x+3)

$\frac{(x^{5} + 243)}{(x + 3)}$

Step 1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$3$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$243$

Step 2

Divide the highest order term in the dividend $x^{5}$ by the highest order term in divisor $x$ .

$x^{4}$

$x$

$3$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$243$

Step 3

Multiply the new quotient term by the divisor.

$x^{4}$

$x$

$3$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$243$

$x^{5}$

$3 x^{4}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{5} + 3 x^{4}$

$x^{4}$

$x$

$3$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$243$

$x^{5}$

$3 x^{4}$

Step 5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{4}$

$x$

$3$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$243$

$x^{5}$

$3 x^{4}$

Step 6

Pull the next terms from the original dividend down into the current dividend.

$x^{4}$

$x$

$3$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$243$

$x^{5}$

$3 x^{4}$

$0 x^{3}$

Step 7

Divide the highest order term in the dividend $- 3 x^{4}$ by the highest order term in divisor $x$ .

$x^{4}$

$3 x^{3}$

$x$

$3$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$243$

$x^{5}$

$3 x^{4}$

$0 x^{3}$

Step 8

Multiply the new quotient term by the divisor.

$x^{4}$

$3 x^{3}$

$x$

$3$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$243$

$x^{5}$

$3 x^{4}$

$0 x^{3}$

$3 x^{4}$

$9 x^{3}$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $- 3 x^{4} - 9 x^{3}$

$x^{4}$

$3 x^{3}$

$x$

$3$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$243$

$x^{5}$

$3 x^{4}$

$0 x^{3}$

$3 x^{4}$

$9 x^{3}$

Step 10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{4}$

$3 x^{3}$

$x$

$3$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$243$

$x^{5}$

$3 x^{4}$

$0 x^{3}$

$3 x^{4}$

$9 x^{3}$

Step 11

Pull the next terms from the original dividend down into the current dividend.

$x^{4}$

$3 x^{3}$

$x$

$3$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$243$

$x^{5}$

$3 x^{4}$

$0 x^{3}$

$3 x^{4}$

$9 x^{3}$

$0 x^{2}$

Step 12

Divide the highest order term in the dividend $9 x^{3}$ by the highest order term in divisor $x$ .

$x^{4}$

$3 x^{3}$

$9 x^{2}$

$x$

$3$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$243$

$x^{5}$

$3 x^{4}$

$0 x^{3}$

$3 x^{4}$

$9 x^{3}$

$0 x^{2}$

Step 13

Multiply the new quotient term by the divisor.

$x^{4}$

$3 x^{3}$

$9 x^{2}$

$x$

$3$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$243$

$x^{5}$

$3 x^{4}$

$0 x^{3}$

$3 x^{4}$

$9 x^{3}$

$0 x^{2}$

$9 x^{3}$

$27 x^{2}$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $9 x^{3} + 27 x^{2}$

$x^{4}$

$3 x^{3}$

$9 x^{2}$

$x$

$3$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$243$

$x^{5}$

$3 x^{4}$

$0 x^{3}$

$3 x^{4}$

$9 x^{3}$

$0 x^{2}$

$9 x^{3}$

$27 x^{2}$

Step 15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{4}$

$3 x^{3}$

$9 x^{2}$

$x$

$3$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$243$

$x^{5}$

$3 x^{4}$

$0 x^{3}$

$3 x^{4}$

$9 x^{3}$

$0 x^{2}$

$9 x^{3}$

$27 x^{2}$

Step 16

Pull the next terms from the original dividend down into the current dividend.

$x^{4}$

$3 x^{3}$

$9 x^{2}$

$x$

$3$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$243$

$x^{5}$

$3 x^{4}$

$0 x^{3}$

$3 x^{4}$

$9 x^{3}$

$0 x^{2}$

$9 x^{3}$

$27 x^{2}$

$0 x$

Step 17

Divide the highest order term in the dividend $- 27 x^{2}$ by the highest order term in divisor $x$ .

$x^{4}$

$3 x^{3}$

$9 x^{2}$

$27 x$

$x$

$3$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$243$

$x^{5}$

$3 x^{4}$

$0 x^{3}$

$3 x^{4}$

$9 x^{3}$

$0 x^{2}$

$9 x^{3}$

$27 x^{2}$

$0 x$

Step 18

Multiply the new quotient term by the divisor.

$x^{4}$

$3 x^{3}$

$9 x^{2}$

$27 x$

$x$

$3$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$243$

$x^{5}$

$3 x^{4}$

$0 x^{3}$

$3 x^{4}$

$9 x^{3}$

$0 x^{2}$

$9 x^{3}$

$27 x^{2}$

$0 x$

$27 x^{2}$

$81 x$

Step 19

The expression needs to be subtracted from the dividend, so change all the signs in $- 27 x^{2} - 81 x$

$x^{4}$

$3 x^{3}$

$9 x^{2}$

$27 x$

$x$

$3$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$243$

$x^{5}$

$3 x^{4}$

$0 x^{3}$

$3 x^{4}$

$9 x^{3}$

$0 x^{2}$

$9 x^{3}$

$27 x^{2}$

$0 x$

$27 x^{2}$

$81 x$

Step 20

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{4}$

$3 x^{3}$

$9 x^{2}$

$27 x$

$x$

$3$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$243$

$x^{5}$

$3 x^{4}$

$0 x^{3}$

$3 x^{4}$

$9 x^{3}$

$0 x^{2}$

$9 x^{3}$

$27 x^{2}$

$0 x$

$27 x^{2}$

$81 x$

Step 21

Pull the next terms from the original dividend down into the current dividend.

$x^{4}$

$3 x^{3}$

$9 x^{2}$

$27 x$

$x$

$3$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$243$

$x^{5}$

$3 x^{4}$

$0 x^{3}$

$3 x^{4}$

$9 x^{3}$

$0 x^{2}$

$9 x^{3}$

$27 x^{2}$

$0 x$

$27 x^{2}$

$81 x$

$243$

Step 22

Divide the highest order term in the dividend $81 x$ by the highest order term in divisor $x$ .

$x^{4}$

$3 x^{3}$

$9 x^{2}$

$27 x$

$81$

$x$

$3$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$243$

$x^{5}$

$3 x^{4}$

$0 x^{3}$

$3 x^{4}$

$9 x^{3}$

$0 x^{2}$

$9 x^{3}$

$27 x^{2}$

$0 x$

$27 x^{2}$

$81 x$

$243$

Step 23

Multiply the new quotient term by the divisor.

$x^{4}$

$3 x^{3}$

$9 x^{2}$

$27 x$

$81$

$x$

$3$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$243$

$x^{5}$

$3 x^{4}$

$0 x^{3}$

$3 x^{4}$

$9 x^{3}$

$0 x^{2}$

$9 x^{3}$

$27 x^{2}$

$0 x$

$27 x^{2}$

$81 x$

$243$

$81 x$

$243$

Step 24

The expression needs to be subtracted from the dividend, so change all the signs in $81 x + 243$

$x^{4}$

$3 x^{3}$

$9 x^{2}$

$27 x$

$81$

$x$

$3$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$243$

$x^{5}$

$3 x^{4}$

$0 x^{3}$

$3 x^{4}$

$9 x^{3}$

$0 x^{2}$

$9 x^{3}$

$27 x^{2}$

$0 x$

$27 x^{2}$

$81 x$

$243$

$81 x$

$243$

Step 25

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{4}$

$3 x^{3}$

$9 x^{2}$

$27 x$

$81$

$x$

$3$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$243$

$x^{5}$

$3 x^{4}$

$0 x^{3}$

$3 x^{4}$

$9 x^{3}$

$0 x^{2}$

$9 x^{3}$

$27 x^{2}$

$0 x$

$27 x^{2}$

$81 x$

$243$

$81 x$

$243$

$0$

Step 26

Since the remander is $0$ , the final answer is the quotient.

$x^{4} - 3 x^{3} + 9 x^{2} - 27 x + 81$

Divide Using Long Polynomial Division (x^5+243)/(x+3)

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