Long division- polynomials? I don't have a teach' or book to explain to me?

Hi,

To do long division, first make sure that the exponents on one of your variables are in descending order. If there is a missing exponent, fill in a + 0(term with missing exponent).

. . . . . . . . . . .______________________

3x³ +2x²y - xy²)9x^4 + 3x³y - 5x²y² + xy³

x exponents are in decreasing order out front: 3,2,1.

x exponents are in decreasing order inside: 4,3,2,1.

To do this division, divide the first term inside the problem by the first term out front.

9x^4

------- = 3x <=This is the first term in the answer.

3x³

. . . . . . . . . . .__3x__________________

3x³ +2x²y - xy²)9x^4 + 3x³y - 5x²y² + xy³

Multiply the 3x in the answer times the entire expression out front. Write the answer underneath the like terms of the problem.

. . . . . . . . . . .__3x__________________

3x³ +2x²y - xy²)9x^4 + 3x³y - 5x²y² + xy³

. . . . . . . . . . ._9x^4 + 6x³y - 3x²y²_

Change all the signs on the bottom line. Then add. Bring down the next term.

. . . . . . . . . . .__3x__________________

3x³ +2x²y - xy²)9x^4 + 3x³y - 5x²y² + xy³

. . . . . . . . . . _-9x^4 - .6x³y+ 3x²y²_

. . . . . . . . . . . . . . . .- 3x³y - 2x²y² + xy³

To do the next step of this division, divide the first term inside at the bottom of the problem by the first term out front.

- 3x³y

--------- = -y <=This is next term in the answer. Put this in answer.

3x³

Multiply the -y in the answer times the entire expression out front. Write the answer underneath the like terms of the problem.

. . . . . . . . . . .__3x__-_y_____________

3x³ +2x²y - xy²)9x^4 + 3x³y - 5x²y² + xy³

. . . . . . . . . . _-9x^4 - .6x³y+ 3x²y²_

. . . . . . . . . . . . . . . .- 3x³y - 2x²y² + xy³

. . . . . . . . . . . . . . . .- 3x³y - 2x²y² + xy³

Change all the signs on the bottom line. Then add.

. . . . . . . . . . .__3x__-_y_____________

3x³ +2x²y - xy²)9x^4 + 3x³y - 5x²y² + xy³

. . . . . . . . . . _-9x^4 - .6x³y+ 3x²y²_

. . . . . . . . . . . . . . . .- 3x³y - 2x²y² + xy³

. . . . . . . . . . . . . . . . .3x³y + 2x²y² - xy³

. . . . . . . . . . . . . . . . ---------------------------

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 0

The answer is 3x - y and the remainder is zero.

. . . . . ____________

a^n + 8)a^2n - a^n - 6

"a" exponents are in decreasing order out front: 1n,0n.

"a" exponents are in decreasing order inside: 2n,1n,0n.

To do this division, divide the first term inside the problem by the first term out front.

a^2n

-------- =

a^n

To do a division, subtract these exponents 2n - n = n.

a^2n

-------- = a^n

a^n

. . . . . _a^n_________

a^n + 8)a^2n - a^n - 6

Multiply the a^n in the answer times the entire expression out front. Write the answer underneath the like terms of the problem.

. . . . . _a^n_________

a^n + 8)a^2n - a^n - 6

. . . . . _a^2n+8a^n_

Change all the signs on the bottom line. Then add. Bring down the next term.

. . . . . _a^n_________

a^n + 8)a^2n - a^n - 6

. . . . ._-a^2n-8a^n_

. . . . . . . . . - 9a^n - 6

To do the next step of this division, divide the first term inside at the bottom of the problem by the first term out front.

-9a^n

--------- = - 9 <=This is next term in the answer. Put - 9 in answer.

a^n

. . . . . _a^n__-_9____

a^n + 8)a^2n - a^n - 6

. . . . ._-a^2n-8a^n_

. . . . . . . . . - 9a^n - 6

Multiply the -9 in the answer times the entire expression out front. Write the answer underneath the like terms of the problem.

. . . . . _a^n__-_9____

a^n + 8)a^2n - a^n - 6

. . . . ._-a^2n-8a^n_

. . . . . . . . . - 9a^n - 6

. . . . . . . . _- 9a^n -72_

Change all the signs on the bottom line. Then add.

. . . . . _a^n__-_9____

a^n + 8)a^2n - a^n - 6

. . . . ._-a^2n-8a^n_

. . . . . . . . . - 9a^n - 6

. . . . . . . . .__9a^n+72_

. . . . . . . . . . . . . . .66 <=remainder

The answer is a^n - 9 with a remainder of 66.

The answer could also be a^n - 9 + 66/(a^n + 8)

I hope that helps!! :-)

For solving these problems, you should arrange the terms of both the divisor and the dividend in such a way that higher power term comes first. Power of a term means sum of powers of the variables in the term. As an example, consider the term 3x^3 y. This is same as 3x^3 y1. x and y are variables. Power over x=3 and power over y=1. So total power of the term = 3+1 = 4

If two terms have equal powers, then choose any variable. Let us choose variable x. Then first put the term with higher power of x. Do this with both dividend and divisor.

1) (9x ^4 + 3x^3 y - 5x^2 y^2 + xy^3)

--------------------------------------...

(3x^3 + 2x^2 y - x y^2)

Here dividend is 9x ^4 + 3x^3 y - 5x^2 y^2 + xy^3 and divisor is 3x^3 + 2x^2 y - x y^2. The terms are already arranged in the order described above.

We will solve as below. The explanation is given after the solution.

...............................3x - y

...............................___________________________

3x^3 + 2x^2 y - x y^2)9x ^4 + 3x^3 y - 5x^2 y^2 + xy^3

...............................9x ^4 + 6x^3 y - 3x^2 y^2

.............................._........_...........+

...............................___________________________

........................................_3 x^3 y _ 2x^2 y^2 + xy^3

........................................_3 x^3 y _ 2x^2 y^2 + xy^3

........................................+...........+..............-

...................................__________________________

.................................................0

Ans: 3x - y

Put the divisor on the left, then put ), then put dividend. Write result at the top.

See the first term of divisor and also the first term of dividend. First term of divisor is 3x^3 and first term of dividend is 9x^4. If you divide 9x^4 by 3x^3, what will you get? You will get 3x because (9x^4)/(3x^3) = 3x.

So write 3x at the top. Then multiply each term of the divisor by 3x and write the result of multiplication below the dividend. From the dividend, subtract the result of multiplication. Whatever is the result of subtraction, treat that as the new dividend and follow the above steps.

Stop when either the dividend becomes zero or when division cannot be carried.

Try solving the second problem yourself. Then you will understand better. Let me know if you need any further help.

Sorry the text in the solution does not appear properly. But hope explanation will help.

There are different ways to approach this problem. Mainly, cancel out as many common factor as possible, sometimes one at a time. For the first problem x is common factor, so first cancel it, then examine the reduced polynomial for possibility of further simplification.

Cancelling x

(9x ^3 + 3x^2 y - 5x y^2 + y^3)

--------------------------------------...

(3x^2 + 2x y - y^2)

The denominator can be factored as (3x-y)(x+y), so check if the numerator can be written in terms of one of this factors.

The numerator is:

9x^3+3x^2y - 5xy^2 + y^3=9x^3+9x^2y - 6x^2y -6xy^2 + xy^2 +y^3

=(x+y)(9x^2 -6xy + y^2)=(x+y)(3x-y)(3x-y)

numerator/denominator=3x-y

The second problem is little different. Try to arrange the numerator in terms of (a^n +8), the last term will yield the remainder.

numerator=a^2n + 8a^n -9a^n -9 +3=(a^n+8)(a^n-9)+3

the polynomial=[(a^n+8)(a^n-9)+3]/(a^n+8)=(a^n - 9) +(3/(a^n+8))

you have quotient=a^n-9 and the remainder is 3.

Factorise, then simplify --->(3x-y)

long division = (a^n - 9) + 66/(a^n + 8)