how do you work long divison and come up with the remainer for the following algebra problem?

Question

5x+4/-15x^3-32x^2+4x+16

Answer 1

Hey April.

 

Long division with polynomials follows the same old methodology as long division with constants, namely: Divide, Multiply, Subtract, Bring Down. Just like we learned in elementary school.

 

It just gets a little more complicated at the "Multiply, Subtract" steps with polynomials.

 

Let's consider your division:

 

              _______________________

(5x + 4) | (-15x^3 - 32x^2 + 4x + 16)

 

Take a look at your first terms in each polynomial. It's 5x in this divisor, and -15x^3 in the dividend. Your job is to figure out how many times 5x goes into -15x^3. In this case, the answer is -3x^2. You've just done your "Divide" step.

 

              _ -3x^2_________________
(5x + 4) | (-15x^3 - 32x^2 + 4x + 16)

 

Now comes the tricky part, the Multiply. That (-3x^2) term has to multiply BOTH terms in your polynomial, so distribute it to both the 5x AND the 4. Place the result under your dividend as you would in normal long division.

 

              _ -3x^2_________________
(5x + 4) | (-15x^3 - 32x^2 + 4x + 16)

               (-15x^3 - 12x^2)

 

The next step is to Subtract. The easiest way to do this is to flip the signs of the result of your multiplication. If it was first a negative, make it positive; if it was first a positive, make it negative. In this particular case, both terms were negative, so we're going to make them positive. Execute the addition.

 

              _ -3x^2_________________
(5x + 4) | (-15x^3 - 32x^2 + 4x + 16)
                +15x^3 + 12x^2

              =   0x^3 - 20x^2

 

The x^3 term went to zero, which is what we want it to do. The last step is to Bring Down the next term, just like with conventional long division. The term you'll be bringing down is +4x.

 

              __-3x^2_________________
(5x + 4) | (-15x^3 - 32x^2 + 4x + 16)
                +15x^3 + 12x^2
               =   0x^3 -  20x^2 + 4x

 

Now do the same thing you did in the first step. Take a look at the first terms in your polynomials: 5x and -20x^2. Ask yourself how many times 5x goes into -20x^2. The answer is -4x. Distribute this to both terms in your divisor, flip the signs for the subtraction, bring down the 16, and complete your division. The remaining steps are shown below all as one. I placed negative signs outside the parentheses of the multiplication results to indicate flipping the signs (i.e. subtracting).

 

 

              __-3x^2 - 4x + 4_________________
(5x + 4) | (-15x^3 - 32x^2 + 4x + 16)
                +15x^3 + 12x^2
               =   0x^3 -  20x^2 + 4x

                           -(-  20x^2 - 16x)

                        =       0x^2 + 20x + 16

                                           -(20x + 16)

                                        = 0

 

Your remainder in this problem is 0. 

 

Hope this helps.