TAGALOG: Division of Polynomials - Long Division and Synthetic Division #TeacherA

Summary

In this educational video, Teacher A introduces lesson 99 on the division of polynomials, focusing on the long division and synthetic division methods. The video provides a step-by-step guide to dividing polynomials, starting with arranging terms in descending order of powers. Example problems are solved to demonstrate each method, showing how to divide polynomials like 'x^2 - 2x - 8' by 'x - 4' using both techniques. The video concludes with a call to action for viewers to join Teacher A's community for more educational content.

Takeaways

• 📚 The lesson focuses on division of polynomials, specifically using long division and synthetic division methods.
• 🔢 The first example demonstrates long division of the polynomial x^2 - 2x - 8 by x - 4, resulting in the quotient x + 2.
• 📉 In the long division process, the first step is to arrange the dividend in descending order of powers, followed by the division symbol and the divisor.
• ➗ The division of the first terms (leading coefficients) is performed, and the result is multiplied by the divisor and subtracted from the dividend.
• 🔄 This process of dividing, multiplying, and subtracting is repeated until the degree of the remaining polynomial is less than the divisor.
• 📈 Synthetic division is introduced as an alternative method for dividing polynomials, particularly useful for finding specific roots.
• 📝 For synthetic division, coefficients of the dividend are aligned, and the divisor's root is used to successively calculate new coefficients and the remainder.
• 🔎 The second example illustrates the long division of 3x^3 + 7x^2 + 3x + 2 by x + 2, yielding a quotient of 3x^2 + x + 1 and a remainder of 0.
• 🔄 Similar to the first example, the synthetic division for the second polynomial also results in a quotient of 3x^2 + x + 1 with no remainder.
• 👨‍🏫 The instructor, Teacher A, invites viewers to join the community for updates on more educational content.

Q & A

What is the main topic of the lesson presented in the transcript?

The main topic of the lesson is the division of polynomials, specifically focusing on the long division method and synthetic division.

What is the first example given in the lesson to demonstrate the long division method?

The first example is the division of the polynomial x^2 - 2x - 8 by x - 4.

How does the teacher instruct to start the long division of polynomials?

The teacher instructs to start by arranging the dividend in descending order of powers and then dividing the first terms of the polynomial.

What is the role of the divisor in the long division method as described in the transcript?

The divisor is used to divide the first term of the dividend, and then it is multiplied by the result to be subtracted from the next term in the dividend.

What is the result of the first example using the long division method?

The result of the first example using the long division method is x + 2.

How does synthetic division differ from the long division method as explained in the transcript?

Synthetic division is a shortcut method for dividing polynomials that involves fewer steps and is used when the divisor is of the form x - c, where c is a constant.

What is the significance of the divisor being x - 4 in the synthetic division example?

The significance is that it allows for the use of synthetic division, which is more efficient than long division when the divisor is linear and of the form x - c.

What is the process for performing synthetic division as described in the transcript?

The process involves arranging the coefficients of the dividend in descending order, using the zero of the divisor to find the remainder, and then performing a series of multiplications and additions to find the coefficients of the quotient.

What is the result of the first example using synthetic division?

The result of the first example using synthetic division is also x + 2, which matches the result obtained by the long division method.

What is the second example given in the lesson to demonstrate the long division method?

The second example is the division of the polynomial 3x^3 + 7x^2 + 3x + 2 by x + 2.

How does the teacher ensure that the subtraction of polynomials is correctly performed in the long division method?

The teacher ensures correct subtraction by emphasizing the importance of changing the sign of the terms when they are subtracted and following the rules of polynomial subtraction.

Outlines

00:00

📘 Polynomial Long Division

The paragraph introduces a lesson on polynomial division, specifically focusing on the long division method and synthetic division. The instructor, Teacher A, begins with an example of dividing a quadratic polynomial x^2 - 2x - 8 by a linear polynomial x - 4. The process involves arranging the dividend in descending order of powers and then performing the division step by step, multiplying the divisor with the result of each division to subtract from the dividend. The instructor demonstrates how to simplify the expression and obtain a quotient of x + 2. This method is essential for solving problems in algebra and understanding polynomial division.

05:00

🔄 Synthetic Division Explained

In this segment, the instructor explains synthetic division, another method for dividing polynomials. Using the same polynomial x^2 - 2x - 8 divided by x - 4, the process starts by arranging the polynomial in descending order and then setting up the division with the divisor. The synthetic division algorithm simplifies the process by using a shortcut method that involves fewer steps than long division. The instructor walks through the steps, including bringing down the first coefficient, multiplying by the root, and adding to get the next coefficient. The final result is the same quotient of x + 2, confirming the method's accuracy.

10:04

📐 Further Exploration of Polynomial Division

The third paragraph continues the exploration of polynomial division with a new example: dividing the cubic polynomial 3x^3 + 7x^2 + 3x + 2 by the linear polynomial x + 2. The instructor demonstrates the long division method, explaining each step in detail, including dividing the leading terms, multiplying the divisor by the result, and subtracting from the current polynomial. The process is repeated until the remainder is zero or the degree of the remainder is less than the divisor. The instructor concludes with the quotient 3x^2 + x + 1, showcasing the method's application to higher-degree polynomials.

15:05

🔄 Synthetic Division for Cubic Polynomials

The final paragraph revisits synthetic division, this time applying it to the cubic polynomial 3x^3 + 7x^2 + 3x + 2 divided by x + 2. The instructor guides through the process, emphasizing the arrangement of coefficients and the use of the divisor's root to perform the division. The synthetic division is shown to be an efficient way to find the quotient, which matches the result obtained through long division. The instructor concludes by encouraging viewers to subscribe to the community for more educational content, highlighting the importance of understanding different methods for polynomial division.

Keywords

💡Polynomials

Polynomials are algebraic expressions that consist of variables and coefficients, involving only the operations of addition, subtraction, and non-negative integer exponents. In the video, polynomials are the main objects of study, specifically focusing on their division. The script mentions dividing polynomials like 'x squared minus 2x minus 8' by 'x minus 4', illustrating the application of polynomial division in algebra.

💡Long Division Method

The long division method is a procedural technique used to divide one polynomial by another. It is analogous to the long division process used with numbers. The video script provides a step-by-step guide on how to apply this method to polynomials, starting with arranging the dividend in descending order of powers and then dividing and multiplying terms to obtain the quotient and remainder.

💡Synthetic Division

Synthetic division is a more efficient method for dividing polynomials, particularly when the divisor is of the form 'x - c'. It involves fewer steps than long division and is demonstrated in the script as an alternative to long division. The video explains how to set up the coefficients of the dividend and the divisor, perform the division, and arrive at the quotient polynomial.

💡Dividend

In the context of division, the dividend is the expression or number that is being divided. In the video, the dividend is the polynomial that the teacher is demonstrating how to divide using both long division and synthetic division methods. Examples from the script include 'x squared minus 2x minus 8' and '3x cubed plus 7x squared plus 3x plus 2'.

💡Divisor

The divisor is the expression or number by which the dividend is divided. In polynomial division, the divisor is typically a simpler polynomial, such as 'x - 4' or 'x + 2'. The script explains how to use these divisors to divide the given polynomials, demonstrating the process of finding the quotient.

💡Quotient

The quotient is the result of a division operation. In the video, the teacher calculates the quotient when dividing the given polynomials. The script shows how the quotient is derived step by step, first by dividing the leading terms and then continuing with the rest of the polynomial terms until the division is complete.

💡Remainder

In polynomial division, the remainder is what is left over after the division process. The script explains that if the remainder is zero, the divisor is a factor of the dividend. The teacher demonstrates how to find the remainder after performing both long division and synthetic division.

💡Coefficients

Coefficients are the numerical factors in a polynomial that multiply the variables. The video script discusses how to handle coefficients during the division process, especially when setting up for synthetic division, where the coefficients of the dividend are used to calculate the quotient.

💡Exponents

Exponents indicate the number of times a variable is multiplied by itself. In the video, the script mentions arranging the dividend in descending order of the exponents before performing division, which is a crucial step in ensuring the correct application of the division algorithm.

💡Variables

Variables are symbols, often represented by letters, that represent unknown values in algebraic expressions. The video script uses variables like 'x' in polynomials to demonstrate division. The teacher explains how to treat variables and their exponents during the division process.

Highlights

Introduction to lesson 99 on division of polynomials.

Explanation of the long division method for polynomials.

Step-by-step guide to long division with the example of x^2 - 2x - 8 divided by x - 4.

Division of the first terms in the polynomial.

Multiplication step in long division to find the next term.

Subtraction step to simplify the polynomial.

Process of bringing down the next term in the polynomial.

Final answer of the long division example, x + 2.

Introduction to synthetic division method.

Setting up the synthetic division with the same polynomial example.

First step in synthetic division: arranging the dividend.

Second step: using the divisor to find the roots.

Synthetic division process: bringing down and multiplying.

Final result of synthetic division, confirming the quotient x + 2.

Transition to a second example with a different polynomial.

Long division of 3x^3 + 7x^2 + 3x + 2 by x + 2.

Division, multiplication, and subtraction steps in the second example.

Final quotient from the second example, 3x^2 + x + 1.

Synthetic division of the second polynomial example.

Final result of synthetic division for the second example, confirming the quotient.

Encouragement for viewers to join the community for updates.

Transcripts

good morning guys teacher a here and

welcome to guru penaisa america so for

today i'm lesson 99 division of

polynomials so pathology modules the

learning tasks

long division method and synthetic

division

let's have example number one

so i'm given nothing i x squared minus

2x minus 8 divided by x minus four so is

to solve nothing step by step starting

with long division

so panda bayon first step

okay lemon you're adding dividend nasa

descending order of powershot it

disappears

exponents

so we have two

one

and then

variable your constant term so you're

adding given

okay

so after that second step indeed divide

the nothing sha katula divide nothing

multi-digit numbers no elementary title

so

at the division symbol

adding dividend

na

x squared

minus two x

minus eight

and then celebration divisor

minus four

okay

so subtract the dividing polynomials and

got doing is indeed dividing first terms

of first term okay so they divide

so x squared

divided by x

so therefore

x squared divided by x i x sanatini

okay

and then after that after dividing in

multiply

so you will multiply your quotient your

partial quotients adding divisor so x

times x i x squared

katakana in the x squared

and then x times negative four a number

and the letter multiplied

so we have negative four x

and then after that after you multiply

is to subtract so the ms natal so

traffic

automatically

cancel out your first term

okay so

x squared minus x squared that's zero

so negative two minus negative four

so say negative four

this is subtraction your rule in

subtracting polynomials i

young minus

negative two plus positive 4. so

following the rules in adding

is to subtract so 4 minus 2i

2

and then x

positive shock assemblies melaka c4 at

positive c4 okay and then after that

bring down you should do the term which

is negative eight

okay

so repeat the process divide ou let your

first term is a first term

so 2x divided by x so c2x pinangasi x

positive

and then

multiply

two times x making a two x

then two times negative four i negative

eight

and then after multiplication subtract

you're negative by becoming positive

you're positive

negative and then you minus magicking

2x plus negative 2x different signs ema

minus so that's 0. 3 negative 8 plus

positive 8 different signs minus that is

equal to zero

yeah

so don't end i'm adding public divide so

base theta starting solution

i'm at the answer i

x plus two okay so that's the long

division method now each of nothing

number

nothing using synthetic division okay

okay so same given dial x squared minus

two x minus eight divided by x minus

four

so

in dividing polynomials using synthetic

division again your first step not then

i think dividend okay

descending order of powershot meaning

your exponent

so we have two an exponent

one now exponent and then zero i'll say

that x okay so

done

and then

in second nothing

human numerical coefficients now

dividend

so we have

one

and then you sustained a no negative two

and then what's next negative eight so

you're not adding dividend now

i think divisor which is x minus four

here

on third step now then i solve for x

so para mass of x

nothing just a zero

so we have x

minus four

is equal to zero solving for x

x is equal to it transpose is negative

force habit by giving positive four

so therefore you divisor naga gamma t

naught and i four

draw a line there

usually of course you like

equal sign okay

so

on synthetic division

i think dividend

okay

divide

first step

it bring down your first number which is

one

there you go

bring down the tile

next is emo multiply nothing human

and then

one times four the answer is four sun

illa lagai salo

okay don is the next number so we have

here four

now

add

following the rules in adding integers

negative two plus positive four make by

subtract four minus two is two

and that is positive because

foreign okay

now repeat the process multiply again

so two times four i

so we have

zero

okay

so two minus one is one therefore

my ex

squared my x cube my x to the fourth my

x to the fifth and so on and so forth

okay

again from right to left

you do love your remainder

constant you should not give my x so not

my x squared x cube x to the fourth x to

the fifth x to the sixth etc etc okay so

i rewrite nothing though so therefore

we have one

and then casamay x

again hindi no no let's see one okay

next this is positive two so that's plus

two

so therefore since zero remainder

nothing i'm adding final answer i

x plus two

yes so therefore

okay

example number two

number two we have three x plus three x

cubed plus two plus seven x squared

divided by x plus two so again

it's a long division

first step it arranged nothing in

dividend

in descending order of powers

exponent so

exponents are given i you my three so

so we have three

x

cubed

and then after num three two so

plus

seven

x squared

and that's also not i one you make x

lamp

plus three x

and then your constant not two

then divided by x plus two okay so

it divide the nut and sha

we have x plus two salabus

dividend number three x cubed

plus seven x squared

plus three x

plus two

oh my

indica

so first step

process not then e divide first terms

the first term

so three x cubed

divided by x so c x cubed

so therefore this is three x squared so

nothing is three x squared sata s

my x squared then so we have here three

x

squared okay

and then next sub process now then

multiply so three x squared times x so

say three x squared the negative

compression is upon x so name x naught n

so three

x cubes

objective

out okay

and then

three x squared times two so three times

two is six

and then copy against the x squared

so after that after another

multiplication subtraction so you

subtract again sub two subtract you

should change nothing you sign on

six x squared the positive magnitude

negative then

okay

so

this will be cancelled out

oh my god three x cubed plus negative

three x cubed macheba is a trap that's

zero next seven minus six i one

and then x squared

is

next after my subtract bring down and

the basis is not the term a union plus

three x

and then after that just repeat the

process

x cube x sub sorry x squared

tinang

is an x d by k x

so plus x

after d divide and multiply x times x is

an x charge upon x d divided by x na

then x times two that's two x

and then we subtract

so again capacitor drop change the sign

so this will become plus this is minus

or negative this is negative

so x squared plus negative x squared

much about the sine subtract that's zero

and then after that

three x plus negative two x marker

banana sign is subtract three minus two

is one and then cx

again hindi nakailang is

after that it bring down your last term

which is plus two

okay so

so we have x plus 2

so

it divides

x

divided by x so x divided by x

okay one any number or letter multiply

and multiply sorry divided by itself as

i got i one okay

so that's one so plus one here and then

multiply one times x i x

1 times 2 i 2

then you subtract x minus x is 0 2 minus

2 is 0. so therefore

0 and la la basna remainder well

remainder okay

so therefore i'm adding quotient i

it's over there

three x squared plus x plus one okay now

each other using synthetic division

so synthetic division first step the

arrangement given in adding dividend

into

descending order of powers

and then second step is

numerical coefficients

we have three

followed by

seven

and then we have three

and then we have two

so he checked

one two three

and then after starting dividend

prototypes out in divisor on divisor

nothing x plus two so nothing in value

in the x by equating this to zero so x

plus two

is equal to 0

solving for x it transpose it positive

to

so in divisor not in a negative 2 sali

nakahualai

draw a line

bring down the first number which is

three

again

that's addition

following the rules in adding

so seven plus negative six okay we're

gonna sign a minus seven minus six i one

and then repeat the process multiply

again one times negative two i negative

two

and then

add following the rows in adding

integers

subtract three minus two i

one

and then after that it multiplies

so one times negative two i negative two

now

let's add

even the sign you subtract two minus two

is zero okay

next

diagonal line i said that's the

remainder

it's a young constant so from right to

left remainder

constant which means relational letter

prolonged variable

so so now you my x

and this is the given x squared

that was

less than the highest exponential

dividend okay so looking at the answer

here it rearranged

so we have three

x squared

and then plus one x hindi

is

one plus x naught and then plus one okay

so therefore

at

and please if ever you are not yet a

member of my community please subscribe

updated my

latest

by teacher a okay that's all for today

see you in my next video