Dividing Polynomials and The Remainder Theorem Part 1

This lesson shows how to divide a polynomial by a binomial using both long division and synthetic division. The lesson also discusses the Remainder Theorem and shows how to use it to find remainders in algebraic divisions. This is the first part of a three part lesson. This video was created for the MHF4U Advanced Functions course in the province of Ontario, Canada.

[AlRichards]

If a term is missing then in Long Dvision you'd insert the missing term with a coefficient of 0. If the x^4 was missing this would look like 0x^4. In synthetic Division, you simply insert a 0 for the missing term. For example, if the polynomial was 2x^3 - 4x + 5 (so the x^2 term is missing), then in the Synthetic Division you'd use the numbers 2 0 -4 5.
In Long Division you'd wite it as 2x^3 + 0x^2 - 4x + 5.

[AlRichards]

In long division you place the divisor on the left (x - 3 in this case). Synthetic division is really long division without the variables. You just operate on all the coefficients & constants instead. I don't really have a better explanation of why, but it's just the restriction that goes on the left for the synthetic division. You aren't really dividing by something that makes it undefined, since you don't (for e.g.) divide the 2 by the 3, it just a procedure that replicates the long div.

[Mag]

Thanks AIRichards314, you've helped quite a bit, clearing up what was fuzzed in the classroom.

[snehasunil]

oh my god it's from such a long year...on 2009 i was just a toddler now here im finding a old video....it's such a nice video....thanks for it

[AlRichards]

Since DIVIDND = DIVSR XQUOTNT + REM, then solving for the DIVSR = (DIVIDND - REM)/QUOTNT.

[AlRichards]

@Ghestar7 Also, in the first example, you subtract in long division, so it is 15 - 15 = 0 (not 15 + 15).

[AlRichards]

In Long Division the lines are subtracted, not added like in Synthetic Division. So it actually is -8x - (-3x) = -8x + 3x = -5x.

[AlRichards]

@sadafbenaf Since (2x^3 + 8x^2) - (2x^3 + 8x^2) = 0 above the long division you'd place a 0x to the right of the 2x^2 and then instead of bringing down just the 5x you also bring down the + 4. Then 5x divided by the x from the x + 4 is 5, and 5(x + 4) = 5x + 20 so you write 5x + 20 below the 5x + 4 and subtract. (5x + 4) - (5x + 20) = 5x + 4 - 5x - 20 = -16, so the remainder is -16 and the quotient is 2x^2 + 0x + 5 or just 2x^2 + 5. Hope this helps.

[AlRichards]

@thisisnotjustinhi If you do the long division you actually divide by the correct binomial. So, if you are dividing by x - 3 then it is x - 3. In synthetic division the number you place to the left of the division symbol is actually the resrtriction in the division. So if you are dividing by x - 3 then to get the restriction you set x - 3 =/ 0 and solve for x which gives x =/3 so 3 goes on the left side of the division symbol. By the way "=/" means "is not equal to". Hope this helps.

[AlRichards]

The polynomial 2x^3 - 5x^2 - 8x + 15 ends in 15.  This means that the product of all the constants at the ends of all factors must be +15. For example x^2 - 7x + 12 factors into (x - 3)(x - 4). Notice that the -3 and -4 multiply to the +12 at the end of x^2 - 7x + 12.
Now, you can actually divide anything. However, to factor (divide and always get a remainder of zero) you try factors with constants that divide into the constant at the end of the original polynomial.

[MeadeyProductions]

Hey Al, thank you very much for this.
Our teacher showed us the long algebraic division method, like the one on your video, and while checking for a little extra guidance on it I came across this.
I'm glad I did because I now know how to use a much more simple and faster method.
Thanks :)

[AlRichards]

@JanJanco Yes, that is correct. If you get a remainder then it is not a factor. A zero remainder means it is a factor.

[AlRichards]

If there's a remainder of 0 that means it is possible to factor the polynomial and so that's what you would normally do. A remainder of zero means it divides in evenly. Like with numbers 20/5 = 4 and the remainder is 0 so 20 = 5*4, but 20/8 = 2 with a remainder of 4 so 20 = 8*2 + 4. 5 is a factor of 20, but 8 isn't a factor of 20. The same is true of polynomials.

[AlRichards]

@lashayshay01 Factors are always of the form {x - number}. So if +2 is a root or zero, then {x - 2} is a factor. Notice the minus after the x is part of the general from of any factor, not the sign from the number +2. If the root or zero is -2 then the factor that divides evenly is {x - - 2} which simplifies to {x + 2}.

[AlRichards]

@breedencm Thanks for the comment. You are correct. For b(x)/a(x) = q(x) + r/a(x) you do need the restriction r /= 0, but for b(x) = a(x)q(x) + r you do not. I am glad you commented on this, because it will improve the meaning of the lesson.
Thanks again, Al.

[Bus]

Very good and easy to fallow! Yes Please post more!

[AlRichards]

Yes, in synthetic division you add the successive rows.

[eversoris]

Thanks so much for you help AlRichards314. I finally understand what to do :)

[sadafbenaf]

Dear Mr. Al: I highly appreciate your QUICK response to any question we have. Sure enough, your feedback did help. Thank you for being there to help!

[TheRandomperson]

You explained this a lot better than my math teacher