need help solving this

Question

Two polynomials P and D are given. Use either synthetic division or long division to divide P(x)by D(x), and express the quotient P(x)⁄D(x)in the form P(x)D(x) = Q(x) + R(xD(x)

.P(x) = 6x4 − 3x3 + 29x2, D(x) = 3x2 + 13

Answer 1

OK… right off the bat, you’re going to use long division, because unless the dividend D(x) is a linear polynomial (which it is most decidedly NOT), you don’t really have the option of using synthetic division.

So, to do long division, just like we learned it in elementary school, you repeat the same four steps in a loop, over and over:

1.      Divide

2.      Multiply

3.      Subtract

4.      Bring down

The fact that we are doing this with polynomials instead of pure numbers doesn’t change anything.

In this problem, we are dividing

P(x) = 6x^4 − 3x^3 + 29x^2

by

D(x) = 3x^2 + 13

We have:

                                 ___________________________

               3x^2 + 13 | 6x^4 − 3x^3 + 29x^2 (+ 0x) (+ 0)

--> (don’t forget to put in zeroes for the missing powers!)

You’re always looking to knock out the leading term under the division sign (for this iteration, 6x^4), and the means to do that are the leading term in the divisor, 3x^2.

Everyone else is just “along for the ride”.

So… step 1: “What do I have to multiply 3x^2 by to get 6x^4?”

Well… divide 6x^4 by 3x^2 to find out!

6x^4 / 3x^2 = 2x^2

And that’s the first term of our solution:

                                                 2x^2

                                 ___________________________

               3x^2 + 13 | 6x^4 − 3x^3 + 29x^2 (+ 0x) (+ 0)

Step 2: Multiply:

Multiply 2x^2 by (3x^2 + 13)

2x^2(3x^2 + 13)

= 6x^4 + 26x^2

Step 3: Subtract:

What I like to do instead is change the signs and add, since it’s easier to keep straight, and we learned in Algebra 1 that it’s the same thing!

                                                              2x^2

                                 ___________________________

               3x^2 + 13 | 6x^4 − 3x^3 + 29x^2 (+ 0x) (+ 0)

                                  -6x^4             - 26x^2

Step 4: Bring down:

                                                               2x^2

                                 ___________________________

               3x^2 + 13 | 6x^4 − 3x^3 + 29x^2 (+ 0x) (+ 0)

                                  -6x^4             - 26x^2

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

                                              −3x^3 + 3x^2

See how we knocked out the leading 6x^4 term?

That was the whole idea!

****************************************************

Start the process again, this time focusing on −3x^3:

Step 1: Divide

−3x^3 / 3x^2 = -x

Which is the second term of our solution:

                                                               2x^2    - x

                                 ___________________________

               3x^2 + 13 | 6x^4 − 3x^3 + 29x^2 (+ 0x) (+ 0)

                                  -6x^4             - 26x^2

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

                                              −3x^3 + 3x^2

*******************

Complete this loop, and do it once more, and you get:

  R(x) = 13x – 13

Q(x) = 2x^2 - x + 1

I checked the answer, and it works.

Answer 2

Q(x) = 2x² -x +1

R(x) = 13x -13

If you don't get the above answer, please ask for more help.

Note: I would not try to use synthetic division unless you are very familiar with its extension to non-linear divisors.