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

Question

Solve with long division

Answer 1

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.

Answer 2

 

   3x³                     -5

   x-2 / 3x⁴ - 6x³ + 0x² -5x + 10

+ -3x⁴ - +6x³ 

   0       0         - +5x + -10    

      0      0 

1. x goes into 3x⁴, 3x³ 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.

Answer 3

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

Answer 4

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