Factor Theorem and Synthetic Division of Polynomial Functions

This precalculus video tutorial provides a basic introduction into the factor theorem and synthetic division of polynomials.
Introduction to Polynomials: • Polynomials - Classify...
Polynomials - Basic Operations:
• Polynomials - Adding, ...
Dividing Polynomials By Monomials:
• Dividing Polynomials B...
How To Graph Polynomial Functions:
• How To Graph Polynomia...
Intermediate Value Theorem:
• Intermediate Value The...
_____________________________________
Long Division - Polynomials:
• Long Division With Pol...
Polynomial Long Division - More Examples:
• Polynomials - Long Di...
Synthetic Division of Polynomials:
• Synthetic Division of ...
Remainder Theorem & Synthetic Division:
• Remainder Theorem and ...
Factor Theorem & Synthetic Division:
• Factor Theorem and Syn...
Rational Zero Theorem:
• Finding All Zeros of a...
______________________________________
Real & Imaginary Solutions - Polynomials:
• How To Find The Real &...
Solving Polynomial Equations - Synthetic D:
• Solving Polynomial Equ...
Solving Polynomial Equations With Excel:
• Solving Polynomial Equ...
Writing Polynomial Functions Given Zeros:
• Writing Polynomial Fun...
Descartes Rule of Signs:
• Descartes Rule of Signs
Solving Polynomial Inequalities:
• Solving Polynomial Ine...
_____________________________________
Final Exams and Video Playlists:
www.video-tuto...
Full-Length Videos and Worksheets:
/ collections

[@TheOrganicChemistryTutor]

Next Video: kzbin.info/www/bejne/f5LUaK1tp8qtpK8 Final Exams and Video Playlists: www.video-tutor.net/ Functions and Graphs: kzbin.info/www/bejne/kIWop2uFo6yUrtE Transformations of Functions: kzbin.info/www/bejne/qqq0faNsrZJ6pbc Graphing Polynomial Functions: kzbin.info/www/bejne/qICylpmaprKpqqM

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ur learning this in grade 12? Im learning in grade 10, thought this was supp to be lower grade stuff

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[@user-mr3wf3xy4t]

So Remainder Theorem tells us that the remainder is always equal to f(c). And Factor Theorem tells us that if the remainder is 0, then the smaller polynomial is a factor of the other. Is that it? Or no?

[@thef.b.i2705]

Yep that’s right

[@Aetherakabane]

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[@-tweeomoz-1786]

This has got to be the most tedious genre of math we've had to do so far; usually, algebra is so fun, but these problems just drag on. Thanks for the video, really helped with my comprehension :)

[@neillawrence4198]

The last example is not correct. If you multiply (2x+3) times (12x^2 - 2x - 2) as shown in bottom right at about 7:30, you do not get back to the original expression. The factor (2x+3) should be left as (x + 3/2). (x + 3/2) times (12x^2 -2x - 2) = (12x^3 + 16x^2 - 5x -3) the original expression. However, if you ignore and continue to factor, you do end up with the same final three solutions. Back two years later to edit. (12/29/23) To clarify, the equality statement at 7:30 as stated is not correct. Expanding the equality as shown gives a result off by a factor of 2. Eliminating the fraction from (x+3/2) introduces a factor of 2 into things. He later factors this out of the polynomial, then removes completly by dividing both sides of the equality by 2. Where this 2 came from and what happened to it he never explained.

[@yesimkhanh9932]

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[@3lackman761]

You're right

[@aszpnh1538]

LOL HE did explain it. Notice he took out a two when factoring 12x^2 - 2 x -2

[@aszpnh1538]

He basically just multiplied the whole expression by 2 on both sides willing doesn’t change the value. Please understand this.

[@neillawrence4198]

@@aszpnh1538 I do fully understand. What he did to eliminate the fraction by introducing a factor of two, then factoring out later in the final solution is perfectly fine, and I most likely would have done that myself. What I am pointing out is that the equality shown at the bottom at 7:30 IS NOT EQUAL. The left side does not have the factor of two, the right side includes the factor of two. If you foil the right side you do not get the left side, therefore the two cannot be shown as equal. The left side needs to be shown as (2)(12x^3+16x^2-5x-3) in order for the equality to be true. He did say what he was doing in each step, but someone just getting into this may not follow. When he removed the factor of two it would help others if he remarked he was removing the factor of two that he had introduced earlier. OK, nuff said, I'm done.

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[@views-lf9mx]

Correct me if I am wrong but I think you are doing the vieta formula wrong and it is not the first video. For ex 3:20 you say the product of the zeros are -2 and they add to negative one. That is wrong because they add to +1 ( formula x1+x2= -b/a => -(-1)/1=+1 and product x1*x2=c/a => -2/1=-2.) So this means x´2-x-2 is not equal to (x-2)(x+1). The zeros are -2 and 1 and the equation is a product of (x+2)(x-1). Anyway thnx for all the tutorials in general they are really helpful :)

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[@lynnsalee8645]

I was kinda lost at 2:10 - 2:16 when we were factoring the x ( in the factor (x - 3) ) in the whole function, how did we arrive to 1x^2-1x-2 especially the -2 part I was lost. Anybody who understood??🤔🤔

[@souppe9616]

i got lost there too 🥺

[@Sf08321]

@Lynn Salee Essentially, the answers from the synthetic division are: 1, -1 and -2 so you have to substitute the numbers into the equation so you get 1x^2-1x-2 We get -2 from 1-3=-2

[@falgunsharma6508]

@@Sf08321 try it with long division method the answer is different . @souppe i was struck here too.

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Any videos on equating coefficients?

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maybe

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[@reetikarathore56]

For the first example, how did 4x^2 become 1x when factored by x?

[@cb2738]

2:18 Take a closer look; -4 + 3 = -1 Then x^2 becomes x, because we are taking away an x when we divided the entire equation by (x-3) Hope it helps

[@bwelipitiya78]

so how x become -2 in the first example? @@cb2738

[@LONWABO_R9]

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[@jennanyquist6284]

I thought you found the factor of the quotient by finding what multiplied to ac and added to b. Is it different for synthetic division?

[@rangareddys1401]

How do you know that you should take x-3? Why can’t you take other number???

[@tylerschaus9283]

because the question started that 3 was a 0 meaning it is an x intercept or x value of the graph. he already knew that it was divisible by x-3 he was just doing it to show the process. don’t know if u need this anymore but that’s it

[@_Anna_Nass_]

why is 3 a zero but -3/2 called a root? is it the same thing?

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[@johnnoemartinez8712]

Can you please try to solve polynomial function in synthetic division with a reminder is more than 1.

[@clli7943]

best.

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Is there anyone watching it again like me😊

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Best teacher

[@keypo790]

So its like, plugging in different numbers and waiting until zero comes out, so that there we can easily use synthethic division to get the factors of the polynomials, OR by using quadratic formula to find that value of x.

[@keypo790]

cubic equation what i mean

[@Backwithanothermilk-HELP_HELPP]

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[@sk-rz8qr]

if the last factor cannot be factored further for example: x^2-5x+12, what do you do?

[@JatPhenshllem]

The quadratic equation

[@gdoggx1336]

But what would you do if in question 1 they didn’t tell you if 3 was a 0 of f(x). And what would you do if they didn’t tell you -3/2 was your root in the second equation. I’m trying to do this but in my questions they don’t give you a hint or straight out tell you was f(x) is. They just give me a polynomial equal to zero and ask me to factor. Please help, thanks.

[@delinaberhane1963]

Gdogg x13 you graph the equation in desmos I don’t know if you know about desmos. But it is actually a website that you can use to graph.and then you look at the point where it cross the x axis I hope this helps.

[@EdohoEric]

Great video but what happens to the (2) at the end of the video, after equating the rest to zero and getting their respective value for x what happens to the (2) amongst them?

[@ChanithDinayana]

Thank you❤

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[@kerubifimanekapohamba4001]

Solve the following equation for me please using factor theorem..in this situation no factor provided. 1. 2x³-7x²+2x+3=0 2. 6x3+19x²+x-6=0

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10:30 thanks again for teaching

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[@aspher_luminous]

What if the polynomial has a remainder when using the synthetic division, can you still use factor theorem if the sythetic divison used to the polynomial has a remainder????

[@joycearabellacagampang1652]

What if its p(x)= x^4 - 5x^2 + 4 ; x - 2 ?

[@clli7943]

then the x^3 place will be 0 and same with x^1 place

[@sinenhlanhlamthembu6671]

what happened to the common factor 2 after factorising ?

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I don't know he neglected it I think he should have distributed it by multiplying it by each bracket.

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Maybe he just 4got

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thank you bro

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Math Jesus May you give us hope for our upcoming exam

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5:03 the -3/2 would be converted to +3/2

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When solving a factor of theorem you'll use synthetic division???

[@meownamejeff]

why is it called a root instead of just a zero

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the root is basically when it crosses the x-axis which is the same as a zero

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How do you find c?

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The last question is wrong

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