Divide 8x^4 – 6x^3 + 7x^2 – 11x + 10 by 2x – 1 using long division. Show all your work. Then explain if 2x – 1 is a factor of the dividend.

Question

anonymous · Answer

hey @jojo4eva  can u help me ?

whpalmer4 · Answer

---------------------------------------------
2x - 1  | 8x^4-6x^3+7x^2-11x+10

divide the first term of the numerator ($8x^4)$ by the first term of the denominator $(2x)$  What do you get?  That will be the first term of the quotient.

anonymous · Answer

4x^3

whpalmer4 · Answer

sorry, got called away from the computer.

right, $4x^3$ is the first term of the quotient.  Now we multiply that by the denominator (the whole thing), giving us$$4x^3(2x-1) = 8x^4-4x^3$$We subtract that from the numerator:
                        4x^3
          ---------------------------------------------
2x - 1  | 8x^4-6x^3+7x^2-11x+10
         -  (8x^4 -4x^3)
            -------------
                        -2x^3 + 7x^2 - 11x + 10

Now we repeat the process.  What is the first term of the remaining numerator $(-2x^3)$ divided by the first term of the denominator $(2x)$?

whpalmer4 · Answer

$$-x^2(2x-1) = -2x^3+x^2$$We subtract that from the remaining numerator:

                        4x^3 - x^2 
          ---------------------------------------------
2x - 1  | 8x^4-6x^3+7x^2-11x+10
         -  (8x^4 -4x^3)
            -------------
                        -2x^3 + 7x^2 - 11x + 10
                      -(-2x^3 + x^2)
                     ---------------
                                        6x^2 - 11x + 10

we keep doing this procedure until we are left with 0, or a quantity that is not divisible by the denominator.  In the latter case, that is the remainder.

anonymous · Answer

Can you continue @whpalmer4  ?

anonymous · Answer

@kewlgeek555

anonymous · Answer

@johnweldon1993

anonymous · Answer

please help yall im extremely bad at this!

whpalmer4 · Answer

Okay, our remaining polynomial is $6x^2-11x+10$.  First term is $6x^2$.  What is $(x^2)/(2x)$?  Again, first term of remaining polynomial divided by first term of denominator.

anonymous · Answer

1/2 x

whpalmer4 · Answer

$$\frac{6x^2}{2x} = \frac{1}{2}x$$?

whpalmer4 · Answer

sorry, I mistyped a bit, I meant $(6x^2)/(2x)$

anonymous · Answer

i did x^2/2x

anonymous · Answer

3x

whpalmer4 · Answer

Yes, you did.  Serves me right for not making you identify the terms to divide :-)

anonymous · Answer

do i put that above the x term?

whpalmer4 · Answer

Right.  Now multiply that result by the denominator, and subtract that result from the remainder of the numerator, what do you get?

anonymous · Answer

the result of what,

whpalmer4 · Answer

multiply 3x by the denominator.  what do you get?

whpalmer4 · Answer

what is the denominator?

anonymous · Answer

6x^2 -11x+10?

whpalmer4 · Answer

No, the original problem was to divide $8x^4 – 6x^3 + 7x^2 – 11x + 10 \text{ by } 2x – 1$.  Which of those quantities is the denominator (or divisor)?

anonymous · Answer

2x-1

anonymous · Answer

so 6x^2 -3x

whpalmer4 · Answer

Correct.  So, our latest term of the quotient is $3x$.  Multiply $3x$ by the denominator to get the latest polynomial to subtract from what is left (the bottom line in the problem so far)

anonymous · Answer

subtract what i just got from 6x^2 -11x+10?

whpalmer4 · Answer

Yes:


                        4x^3 - x^2 
          ---------------------------------------------
2x - 1  | 8x^4-6x^3+7x^2-11x+10
         -  (8x^4 -4x^3)
            -------------
                        -2x^3 + 7x^2 - 11x + 10
                      -(-2x^3 + x^2)
                     ---------------
                                        6x^2 - 11x + 10

whpalmer4 · Answer

$$6x^2-11x+10 - (6x^2-3x) = $$

anonymous · Answer

10-8x

whpalmer4 · Answer

Good. that gives us:

                        4x^3 - x^2 + 3x
          ---------------------------------------------
2x - 1  | 8x^4-6x^3+7x^2-11x+10
         -  (8x^4 -4x^3)
            -------------
                        -2x^3 + 7x^2 - 11x + 10
                      -(-2x^3 + x^2)
                     ---------------
                                        6x^2 - 11x + 10
                                    - ( 6x^2   -3x)
                                     -------------------
                                                    -8x  + 10

Okay, divide the first term of the remaining part by the first term of the denominator.  What is the result?

anonymous · Answer

divide -8x by 2x-1?

whpalmer4 · Answer

No, the first term of the remaining part, divided by the first term of the denominator.

whpalmer4 · Answer

Same procedure we've used for the previous 3 terms.

anonymous · Answer

oh -11x by 2x-1?

anonymous · Answer

having trouble dividing that

whpalmer4 · Answer

no, divide the first term of the remaining part, which is $-8x+10$ as you can see from the typed rendition above

divide the first term of that by the first term of $2x-1$

anonymous · Answer

i got -4+ 6/2x-1

whpalmer4 · Answer

Yes, so the entire answer is $$4x^3 - x^2 + 3x-4 + \frac{6}{2x-1}$$ or $$4x^3 - x^2 + 3x-4 \text{ with remainder } 6$$

anonymous · Answer

so 2x-1 isn't a factor of the dividend?

whpalmer4 · Answer

nope, a factor will leave a remainder of 0.

anonymous · Answer

okay

anonymous · Answer

thanks for walking me through that

anonymous · Answer

i really appreciate it!

whpalmer4 · Answer

That polynomial $$8x^4-6x^3+7x^2-11x+10$$is irreducible, which is a fancy word for "can't be factored"

anonymous · Answer

okay, thanks

whpalmer4 · Answer

you're welcome!  I think (despite the hassle of typing them) these are actually easier to do than long division on numbers, because you never have to worry about your guess for the next term being "too big".  You also don't have to worry about getting the digits in the right columns, or any of that.  On the other hand, subtracting polynomials with - signs in them is somewhat error-prone for some (myself included).