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# (3x^4-6x^3-5x+10)/x-2

Solve with long division

Polynomial long division is no different than regular long division. We begin by rewriting the original question into a more traditional division format:

(3x^4 – 6x^3 – 5x + 10) / (x – 2)

___________________

x – 2 | 3x^4 – 6x^3 – 5x + 10

Now that we have re-written the problem we can solve the problem by identifying what divisor (x – 2) can be multiplied by to solve:

_3x^3______________

x – 2 | 3x^4 – 6x^3 – 5x + 10

The first step is to find what the divisor can be multiplied by to equal the first part of the dividend (3x^4 – 6x^3). By multiplying x – 2 * 3x^3 we get 3x^4 – 6x^3, which we will subtract from the dividend leaving no remainder:

_3x^3______________

x – 2 | 3x^4 – 6x^3 – 5x + 10

3x^4 – 6x^3

0

We now bring down the remaining portion of the dividend (-5x + 10) and then multiply x – 2 by -5:

_3x^3______- 5________

x – 2 | 3x^4 – 6x^3 – 5x + 10

3x^4 – 6x^3          ↓

0 – 5x + 10

– 5x + 10

0

Once again there is no remainder, therefore the answer is 3x^3 – 5.

For the sake of argument, let us say there was a remainder. The solution would then be 3x^3 – 5 + the remained over the original divisor.

3x3                     -5

x-2 / 3x4 - 6x3 + 0x2 -5x + 10

-3x- +6x

0       0         - +5x + -10

0      0

1. x goes into 3x43x3 times, put the term over the x3 place. Multiply 3x3 by x then by -2. Change the signs then add the terms.
2. since you get 0, divide into the next term. x goes into -5x, -5 times, put the term over the x place. Multiply -5 with x then by -2. Change the signs then add the terms.
3. you get 0 as the remainder so the answer is 3x3-5.

Long division of a polynomial is very similar to numerical long division.  Put the numerator polynomial under the division sign and the denominator on the outside:

_3x^3__-5_________________________

X-2) 3x^4-6x^3-5x+10

3x^4-6x^3

-5x+10

Find an expression to multiply the X on the outside by to equal X^4; this is 3x^3.  Multiply through and subtract from the original polynomial.  The multiplication is shown and subtraction gives zero x^4 and  zero X^3.  Bring down the -5x + 10 just as you would for a long division problem.  Now repeat and find an expression for the x on the outside to equal -5x.  The answer is -5.  Multiply through and subtract and find there is nothing left so the division problem is solved.

Rebecca Muegge

Wilton, Ca

Since the divisor is linear, you can use synthetic division to get it done, which is simpler.

2 | 3 -6 0 -5 10

........6 0  0 -10

---------------------------

##### ....3 0 0 -5 |0

So,

(3x^4-6x^3-5x+10)/(x-2) = 3x^3 - 5