How to Divide Polynomials using Long Division - Polynomials
Summary
TLDRIn this educational video, Trigon demonstrates how to divide polynomials using long division. Starting with dividing 6x^2 - 2x - 28 by 2x + 4, Trigon explains the process step-by-step, emphasizing that it mirrors the long division method used for whole numbers. The video illustrates dividing the leading terms, multiplying, and subtracting to find the partial quotient, which in this case is 3x - 7. Trigon then solves another polynomial division problem, reinforcing the method with a new example. The video is designed to teach viewers the algebraic process in an accessible manner.
Takeaways
- 📚 The video tutorial focuses on dividing polynomials using long division, a method similar to dividing whole numbers but applied to algebraic expressions.
- 🔢 The first example involves dividing (6x^2 - 2x - 28) by (2x + 4), where the leading terms are divided first, followed by multiplication and subtraction steps.
- 👨🏫 The instructor emphasizes the importance of dividing the leading coefficients or terms of the dividend and divisor to find the partial quotient.
- 📉 After dividing, the instructor demonstrates how to multiply the entire divisor by the partial quotient and subtract this product from the original polynomial to find the remainder.
- 🔄 The process is repeated with the new polynomial (result after subtraction) to find subsequent terms of the quotient until the remainder is zero or less than the divisor.
- 📝 The quotient obtained from the division is (3x - 7), indicating that the division process has been correctly followed and the remainder is zero.
- 🔍 In a second problem, the method is applied to (3x^3 - 4x^2 - 7x - 5) divided by (3x - 2), showing the consistency of the long division approach.
- 📈 The tutorial highlights the need to bring down terms from the dividend when the current degree of the remainder is less than the degree of the divisor.
- 📌 The final quotient for the second problem is (x^2 + 2x - 1) with a remainder of (-7), which is expressed as -7/(3x - 2).
- 💡 The video concludes with a reminder to like, subscribe, and enable notifications for updates, encouraging viewer engagement with the channel.
Q & A
What is the main topic discussed in the video?
-The main topic discussed in the video is dividing polynomials using long division.
What is the first polynomial division problem presented in the video?
-The first polynomial division problem is 6x^2 - 2x - 28 divided by 2x + 4.
How does the video explain the process of polynomial long division?
-The video explains that polynomial long division is similar to the long division of whole numbers, but in algebraic form. It involves dividing the leading terms, multiplying, and then subtracting.
What is the partial quotient obtained after the first division step in the first problem?
-The partial quotient obtained after the first division step in the first problem is 3x.
What is the result of the multiplication of the partial quotient 3x by the divisor 2x + 4?
-The result of the multiplication of the partial quotient 3x by the divisor 2x + 4 is 6x^2 + 12x.
What is the remainder after the first subtraction step in the first problem?
-The remainder after the first subtraction step in the first problem is -14x - 28.
What is the final quotient obtained after solving the first polynomial division problem?
-The final quotient obtained after solving the first polynomial division problem is 3x - 7.
How does the video handle the remainder in the division process?
-The video brings down the remainder and continues the division process by dividing the leading term of the new polynomial by the leading term of the divisor, then multiplying and subtracting as before.
What is the second polynomial division problem presented in the video?
-The second polynomial division problem is 3x^3 - 4x^2 - 7x - 5 divided by 3x - 2.
What is the final quotient and remainder obtained after solving the second polynomial division problem?
-The final quotient obtained after solving the second polynomial division problem is x^2 + 2x, and the remainder is -7/(3x - 2).
What is the advice given by the video for those who are new to the channel?
-The video advises those who are new to the channel to like, subscribe, and hit the Bell button to be updated with the latest uploads.
Outlines
📘 Polynomial Long Division Introduction
The video begins with the host, Trigon, introducing the topic of dividing polynomials using long division. The first problem presented is dividing \(6x^2 - 2x - 28\) by \(2x + 4\). The process involves dividing the leading terms of the dividend and divisor, which results in \(3x\) as the partial quotient. The host then demonstrates the steps of multiplying the divisor by the partial quotient and subtracting the result from the dividend. The process is repeated until the remainder is zero, yielding the quotient \(3x - 7\).
📗 Detailed Polynomial Long Division Process
The second paragraph delves deeper into the long division process with a new problem: dividing \(3x^3 - 2x^2 - 7x - 5\) by \(3x - 2\). The host explains how to divide the leading terms, resulting in \(x^2\) as the partial quotient. The video shows the multiplication of the divisor by the partial quotient and the subsequent subtraction from the dividend. The process is repeated, leading to a remainder of \(-7\) over the divisor \(3x - 2\), and the quotient is \(x^2 + 2x - 1\).
📙 Conclusion and Call to Action
In the final paragraph, the host concludes the tutorial on polynomial long division by summarizing the steps and presenting the final answer for the second problem: \(x^2 + 2x - 1\) with a remainder of \(-7/(3x - 2)\). The host encourages viewers to like, subscribe, and turn on notifications for updates, and signs off as 'Teacher Trigon'.
Mindmap
Keywords
💡Polynomials
💡Long Division
💡Dividend
💡Divisor
💡Leading Coefficients
💡Partial Quotient
💡Multiply and Subtract
💡Remainder
💡Quotient
💡Algebraic Form
Highlights
Introduction to dividing polynomials using long division
Explanation of the first problem: 6x^2 - 2x - 28 divided by 2x + 4
Step-by-step guide on dividing the leading terms of the dividend and divisor
Calculation of the partial quotient: 6x^2 divided by 2x equals 3x
Multiplication of the partial quotient by the divisor
Subtraction step in the long division process
Continuation of the division process with the next term
Final quotient derivation: 3x - 7
Emphasis on the remainder being zero in the division
Introduction to the next problem involving 3x^3 divided by 3x
Division of the leading terms resulting in x^2
Multiplication and subtraction steps for the second problem
Derivation of the quotient for the second problem: x^2 + 2x
Explanation of the remainder over the divisor
Final expression of the quotient and remainder
Encouragement for viewers to like, subscribe, and hit the Bell button for updates
Conclusion and sign-off by the presenter
Transcripts
hi guys it's me the Trigon in today's
video we will talk about dividing
polynomials using long division
so without further ado
let's do this topic so what we have here
is the first problem in the later on we
will continue solving another problem
so this is the problem guys
we have
6X Square minus 2x minus 28 divided by
2x plus 4. so this is your dividend
and this is your divisor so basically
guys
um if you don't know how to divide
polynomials using long division it is
the same as the long division that you
have learned from your Elementary days
on how to divide whole numbers okay
it's in algebraic form
so let's start
first thing you need to do is to divide
the leading coefficients of your
dividend and your device sorry
leading terms
okay
the first thing you need to do is to
divide the leading terms of your
dividend and device or at your own
leading terminal
dividend
and this is the leading term of your
divisor so this will become 6X Square
divided by 2x I will use this part
Minima solution or side solution
6 divided by 2
is 3
the next Square divided by X is simply X
and this 3x will serve as the partial
quotient so it will be placed here
we have here 3x
okay now this 3x will be distributed or
multiplied one by one by one by two x
and four so what will happen is that we
have 3x times 2x
that is equal to 6 x squared
and this one at the Domain after 2x for
the man 3x times 4 that is Plus
12 X
after multiplying
we have to subtract so in launcher guys
rotation
we divide
we divide the leading terms
and then after dividing multiply and
then lastly subtract 18 adding rotation
now
a multiple inclusion this is
always enclosed by the parenthesis by
parenthesis
multiply okay so that is 6X squared
minus six x Cooperative Omega 0n
okay
zero and then here you're adding
negative 2x minus 12x it will become
negative 14 x okay
negative two x two x nothing
plus minus
12x
negative 2X
so you have negative 14x
then after that you will bring down
negative 28 so negative
28. yeah what's next
so same rotation
divide dial divide nothing in background
leading term which is negative 14x by
the leading term of your divisor
okay so we have negative 14 I will use
this part
negative 14
x divided by
2x so negative 14 divided by 2 is
negative 7 and then X over X is zero I
one so negative seven so this will
become minus
7.
okay so what we have here is the number
one after dividing multiply
negative seven times two x that is
negative
negative
14 x
negative 7
times 4 that is negative
Twenty Eight
so multiplying that then I'll subtract
nothing is
so as you can see
negative 14x plus 14x positive DNA so
zero zero negative 28 plus positive 28
that is zero so plug between zero then
your remainder is zero again your
remainder zero and the quotient
is this
the whole sent
Ence the quotient
is equal to 3x
minus seven or
is that we have 6X Square
minus 2X
minus 28
divided by
2x Plus 4. is equal to 3x
minus seven and that's it guys so that's
something new solving the next problem
okay let's have another problem
for the next problem same rotation
divide the leading terms multiply and
then subtract let's try
okay so let's try this one
eating term leading term 3x cubed
divided by
3x
solution 3x cubed
divided by 3x
so 3 over 3 is 1 x squared over X cubed
over X is x squared
so is x squared
so what's next sir Alexa
multiply one by one
okay so multiplying that then it will
become this one
x square times 3x that is 3
x cubed
x squared minus times negative 2x that
is negative
2x squared so subtract nothing
and don't forget
nah I'm not sure
inclusions
subtraction or negativity positive
so what will happen
is that we have not 3x cubed plus
negative three x again zero negative
zero zero
don't mind 4x squared plus
positive 2x squared that is equal to 6 x
squared then bring down by this is
negative 7X
leading coefficient more 6X Square
divided by 3x so 6 x squared
divided by 3x
so 6 divided by 3 is 2 x squared divided
by X is X so Tanisha Plus
2X
multiply 2x times 3x is 6X squared
2x times negative 2 is minus
4 x again subtract then enclose by
parenthesis
change time operation Plus
so what happened
6X squared plus negative 6X squared
again 0 again guys negative 7x plus
positive 7x that is positive 4X that is
negative 3x then bring down it on last
one then which is negative five
divided let you add in leading
coefficient
we have decimeters apart data
we have
negative
three x over
3x that is definitely
negative one
minus 1.
okay so what will happen is this
multiply that negative times negative 1
times 3x that is
negative 3x
negative one
times negative 2 that is positive 2.
okay
in close by parenthesis
operation
Plus
negative
so what will happen is this
negative 3x plus 3x that is zero zero n
is zero
negative 5 plus
okay plus
negative that is negative seven it is
quotient
tapos
positive negative remainder
over
divisor
so to express our final answer guys
okay
uh divisor
three
x cubed
plus 4 x squared
minus 7X
minus five
divided by
3x minus 2.
is equal to it quotient
in quotient is
x square
plus 2X because
minus one
is negative
negative seven over your divisor
that is 3x
minus 2. so again
quotient Theta
since negative unit remainder
minus seven over your device or which is
3x minus 2. so I hope guys you learned
something from this video on how to do
the long division
in dividing polynomials so if you're new
to my channel don't forget to like And
subscribe but hit the Bell button for
you to be updated latest uploads again
it's me teacher gone
foreign
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