DIVISION OF POLYNOMIALS USING LONG DIVISION || GRADE 10 MATHEMATICS Q1

Summary

This instructional video script outlines the process of dividing polynomials using long division. It guides viewers through each step, starting with arranging polynomials by decreasing exponents and inserting zeros for missing terms. The tutorial demonstrates dividing the first term, multiplying the partial quotient by the divisor, subtracting from the dividend, and repeating until complete. Examples are provided, including dividing \(x^3 - 4x^2 + 3x - 6\) by \(x - 2\), and \(3x^5 + 3x^4 - x^3 + 3x^2\) by \(x^2 + 1\), with each step explained in detail. The script concludes with a prompt to like, subscribe, and stay updated for more educational content.

Takeaways

• 📚 Long division is a method used to divide polynomials.
• 🔢 The process involves arranging the dividend and divisor in decreasing order of exponents, and inserting zeros where necessary.
• ✅ Begin by dividing the first term of the dividend by the first term of the divisor.
• 🔄 Multiply the partial quotient by the divisor and subtract the result from the dividend.
• 🔽 Bring down the next term in the dividend and repeat the process until all terms have been processed.
• 📉 An example given in the script is dividing x^3 - 4x^2 + 3x - 6 by x - 2.
• 🔗 Each step in the division process is explained with a focus on exponents and coefficients.
• 📈 The script demonstrates how to handle negative terms and remainders in polynomial division.
• 📝 The final answer includes the quotient and the remainder, if any.
• 🎓 The video aims to educate viewers on polynomial division, encouraging practice and further learning.
• 👍 The presenter invites viewers to like, subscribe, and stay updated for more educational content.

Q & A

What is the first step in polynomial long division?

The first step is to arrange the dividend and divisor in decreasing power of exponents and insert zeros as coefficients for missing terms if necessary.

How do you divide the first term of the dividend by the first term of the divisor?

You divide the highest power term of the dividend by the highest power term of the divisor using exponent subtraction. For example, in the division of x^3 by x, the result is x^2.

What do you do after dividing the first terms in polynomial long division?

After dividing, you multiply the partial quotient by the entire divisor, and then subtract the result from the dividend.

Why is it important to subtract the results carefully in polynomial long division?

Subtracting the results carefully is essential because any miscalculation will affect the accuracy of the remaining steps. It ensures that the new dividend is correct for the next step.

What happens if you encounter a missing term during polynomial division?

If a term is missing in the polynomial, you must insert a zero as its coefficient to maintain consistency in the process of long division.

How do you handle remainders in polynomial long division?

The remainder, if any, is expressed as a fraction with the remainder divided by the original divisor. For example, if the remainder is -8 and the divisor is x - 2, the remainder is written as -8/(x - 2).

What is the result of dividing x^3 - 4x^2 + 3x - 6 by x - 2?

The quotient is x^2 - 2x - 1, and the remainder is -8. The final answer is x^2 - 2x - 1 + (-8)/(x - 2).

What does 'bring down the next term' mean in polynomial division?

After each subtraction step, you bring down the next term from the dividend and continue the division process until all terms have been processed.

Why do you subtract the exponents when dividing terms in polynomial long division?

You subtract the exponents because of the rules of exponents. When dividing like terms, you subtract the exponent of the divisor from the exponent of the dividend.

How do you handle division when the divisor has multiple terms, such as in 'divide by x^2 + 1'?

In this case, you divide the first term of the dividend by the first term of the divisor, perform the multiplication of the quotient by the entire divisor, and continue the division process with the remaining terms.

Outlines

00:00

📚 Polynomial Long Division Basics

This paragraph introduces the process of dividing polynomials using long division. It outlines the steps: arranging polynomials by decreasing exponent, inserting zeros for missing terms, dividing the first term of the dividend by the first term of the divisor, multiplying the partial quotient by the divisor, subtracting the result from the dividend, and repeating the process until the division is complete. An example is given where the polynomial x^3 - 4x^2 + 3x - 6 is divided by x - 2, demonstrating each step in detail, including multiplying the partial quotient and subtracting from the dividend.

05:03

🔢 Detailed Polynomial Division Example

This paragraph continues the explanation of polynomial division with a focus on a specific example. It details the step-by-step process of dividing x^3 + 8 by 3x + 2. The explanation includes dividing the leading terms, multiplying the divisor by the partial quotient, subtracting the result from the dividend, and bringing down the next term. The process is repeated until the remainder is found. The final quotient is 9x^2 - 6x with a remainder of 4, showcasing how to handle each term and exponent in the division.

10:04

📘 Advanced Polynomial Division with Remainders

The final paragraph presents an advanced example of polynomial division, dividing 3x^5 + 3x^4 - x^3 + 3x^2 by x^2 + 1. It explains how to handle higher degree terms and demonstrates the division process, including bringing down terms and dealing with remainders. The explanation walks through each step, showing how to divide the leading terms, multiply, subtract, and continue the process until the final quotient and remainder are determined. The video concludes with a reminder to like, subscribe, and stay updated for more tutorial videos.

Keywords

💡Polynomials

A polynomial is a mathematical expression consisting of variables, coefficients, and exponents, combined using addition, subtraction, and multiplication. In the video, polynomials are the main mathematical objects being divided using long division. For example, the polynomial x^3 - 4x^2 + 3x - 6 is divided by x - 2 in the script.

💡Long division

Long division is a method used for dividing larger numbers or expressions. In the context of this video, it's applied to polynomials. The process involves dividing the highest degree term of the dividend by the highest degree term of the divisor and continuing step by step. For instance, the video demonstrates dividing x^3 - 4x^2 + 3x - 6 by x - 2 using this technique.

💡Dividend

The dividend is the polynomial being divided in the division process. In the video, the dividend is represented by expressions like x^3 - 4x^2 + 3x - 6. The video explains that to begin the division, you divide the first term of the dividend by the first term of the divisor.

💡Divisor

The divisor is the polynomial that divides the dividend. For example, in the video, x - 2 is the divisor when dividing the polynomial x^3 - 4x^2 + 3x - 6. The process of long division relies on dividing the terms of the dividend by the terms of the divisor sequentially.

💡Exponent

An exponent indicates the number of times a number or variable is multiplied by itself. In polynomials, exponents show the degree of each term. For example, in the term x^3, the exponent is 3. The video emphasizes that when dividing polynomials, the exponents of the terms must be subtracted according to the division rules.

💡Partial quotient

A partial quotient is an intermediate result in the process of long division. Each time a term of the dividend is divided by the divisor, it gives a partial quotient. In the video, after dividing the first term of the dividend by the divisor, a partial quotient (e.g., x^2) is obtained, which is then multiplied back by the divisor.

💡Remainder

The remainder is the leftover part of the dividend that cannot be divided further by the divisor. In the script, after completing the division of x^3 - 4x^2 + 3x - 6 by x - 2, a remainder of -8 is obtained, which is expressed as part of the final answer.

💡Multiplication of polynomials

Multiplying polynomials involves multiplying each term of one polynomial by each term of another, then combining like terms. The video demonstrates this process when multiplying the partial quotient by the divisor, such as multiplying x^2 by x - 2 to get x^3 - 2x^2.

💡Subtraction in long division

In long division, after multiplying the partial quotient by the divisor, the result is subtracted from the current dividend. This step reduces the degree of the polynomial being divided. In the video, for example, x^3 - x^3 gives 0, and -4x^2 minus -2x^2 gives -2x^2.

💡Bringing down the next term

This step in polynomial long division involves bringing down the next term of the dividend after each subtraction. It allows the division process to continue. In the video, after subtracting the terms, the next term (such as 3x or -6) is brought down to continue dividing.

Highlights

Introduction to dividing polynomials using long division

Step-by-step guide on arranging polynomials for long division

Inserting zeros as coefficients for missing terms

Dividing the first term of the dividend by the first term of the divisor

Multiplying the partial quotient by the divisor

Subtracting the result from the dividend

Bringing down the next term in the dividend

Repeating the process until the division is complete

Example division of x^3 - 4x^2 + 3x - 6 by x - 2

Calculating the first term of the quotient

Multiplying and subtracting to find the next term of the quotient

Continuing the division process with the remaining terms

Final quotient and remainder after completing the division

Dividing 2x^3 + 8 by 3x + 2

Calculating the quotient for the second example

Finding the remainder after the division is complete

Dividing 3x^5 + 3x^4 - x^3 + 3x^2 by x^2 + 1

Final quotient and remainder for the third polynomial division example

Encouragement to like, subscribe, and hit the bell button for more tutorials

Transcripts

[Music]

on how to divide polynomials using long

division

okay into your most steps in dividing

polynomials using long division

una we need to arrange the dividend and

the divisor in decreasing power of

exponent

take note insert zeros as coefficient of

the missing terms

of each polynomial if necessary

number two divide the first term of the

dividend by the

first term of the divisor number three

multiply the partial quotient to the

divisor

number four subtract the result from the

dividend and for step number five

bring down the next term in the dividend

and last

repeat the process until done

okay so let's have an example

divide x cubed minus four x squared plus

three

x minus six by x minus two

so gamma two mass steps nian so

paramagne guided diode supports

to solve non-polynomial using the long

division

so

term so therefore we can proceed now in

dividing the

polynomials by x minus two so

antagonitis a

low that is the dividend at x minus to

the magnionian divisor

okay an impang step divide the first

term

of the dividend by the first term of the

divisor so you

divide down at n so using the loss of

exponent

x cubed minus x d divided

by same base exponent so

no subtract not n so three minus one

so my one total unless a denominator not

n so 3 minus

1 the answer is 2 therefore x cubed

divide

x the answer is x squared

and after that so step number 3 we need

to multiply the partial

quotient to that divisor so partial

palancas

indi patapos okay so you multiply

nothing see x

squared okay x minus two and that is

x cubed minus two x squared pattern

x squared times x so multiplying them

entire with the same base

negative exponent

dividing subtracts

exponents so two plus 1 d

exponentially x so that is 2 plus 1 k x

cubed then x squared times negative 2

that is

negative 2 x squared and after that and

the next step

subtract the result from the dividend

so x cubed minus

x cubed minus x cubed the answer is zero

so indiana nothing illegal

next negative four x squared minus

negative two x squared so the answer is

so negative four x squared minus

negative two x squared divided by giving

plus two because negative times negative

positive

so negative 4x squared plus 2x squared

the answer is

negative 2x squared and then after that

bring down positive 3x okay

so udayosa

so negative 2x squared divide x so copy

negative two

and then x squared divide x the answer

is

x bucket two minus one so one eons

exponent now one gen so therefore

negative 2

x squared divide x the answer is

negative

2 x and then an unknown step number to

natalia so

negative 2x multiply to x

minus 2 so negative 2x times x the

answer is

negative 2x squared negative 2x times

negative 2 that is positive for x

and then step number four tile so step

number four

subtract the result from the dividend so

two x squared minus two x squared answer

is zero so in the internet in in

three x minus four x so 3x minus 4x

is negative x so that is negative x

and then step number five bring down the

next term and that

is negative six then uh

step number two so divide

let negative x divide x the answer is

negative one

and then multiply the partial cos idea

multiply negative one to x minus two so

negative 1 times

x that is negative x negative 1 times

negative 2 positive 2

and then subtract the result from the

dividend so

negative x minus negative x the answer

is zero

negative six plus two the answer uh

negative six minus two the answer is

negative eight

okay so maritime remainder and the

negative

eight so therefore panel native nila

gayam final answer net and so

the quotient is x squared minus two

x minus one plus many time remainder

plus negative eight over x minus

final answer next

divide two x cubed plus eight by three x

plus two

okay

so 27 x cubed plus eight

so your highest degree nothing is three

and then your constant term is eight so

sing

term and dividend don't suppress

terminal divisor

27x cubed divide 3x the answer is

so 27 divided 3 that is 9

x cubed mine divide x that is

subtract the exponent so 3 minus 1 that

is x squared

so therefore 27 x cubed divide 3x the

answer is

9x squared and then multiply 9x squared

to 3x plus 2 so 9x squared times 3x that

is

x cubed nine x squared times two that is

eighteen x squared

and then subtract so zero nine

so zero x squared minus eighteen x

squared answer

is negative eighteen x squared and then

after that

bring down zero x okay then repeat the

time sub process so divide that

in on it so repetitive step number two

negative 18 x squared divide three x the

answer is

negative six x okay and then

step number three multiply negative six

x to three x plus two

so negative six x times three x that is

negative eighteen x squared negative 6

x times 2 that is negative 12 x

and then step number 4 subtract so 18 x

squared minus

negative 18 x squared that is zero and

then

zero minus negative 12 the answer is

since 0 x minus negative 12 x

it will become positive so therefore

that is 12x

in step number 5 bring down the next

term and that is positive 8.

so 12 x plus eight then

would it is step number two so divide

twelve

x divide three x the answer is four so

plus four

and then multiply four to three x plus

two

so 4 times 3x that is 12x 4 times 2 that

is 8 so

nothing and the answer is 0. so

therefore the answer

is 9 x squared minus 6 x

plus 4 volatile remainder okay

okay example number 3 divide 3x cubed

plus 3x squared minus

one plus three x to the fifth minus two

x to the fourth

by x squared plus one okay nothing young

even a dividend hindi naka arranged

okay soda patty arranged nothing yen

so that money and three x to the fifth

and then

so soon any negative two x to the fourth

plus three x cubed

plus three x squared and then we'll

attain

the green one

first term and dividends of first term

nondivisor

so 3x to the fifth divide x squared so

copy three

x to the fifth divide x squared that is

three x

cubed because five minus two that is

three so therefore

3x to the 5th divide x squared answer is

3x cubed and then after that multiply

3x cubed times x squared that is 3x to

the 5th

and 3x cubed plus 1 times 1 that is

three x cubed

uh in the fourth power bucket class

exponent since 3 x cube

times 1 is 3 x cubed not in it

exponent and after that subtract

0

that is negative two x four so you bring

down nathanian

negative two x to the fourth and then

you

if we bring down out the young next term

and that is positive three x squared

so a little nice step so step number two

and onoga when

divide the first term of the dividend

sub first term and divisor so negative

2x to the fourth divide

x squared so the answer is negative

two x squared okay can say four minus

two that is two

so the answer is negative 2 x squared

and then after

that negative 2 x squared times x

squared plus 1

negative 2 x squared times x squared is

negative 2x to the fourth

and negative 2x squared times 1 is

negative 2x squared so indeed nothing it

apathy to chi 3 x cube d to nothing

3 x squared okay and then subtract

so it only gives you a nail this

so three x squared minus negative two x

squared so panang are in a five

three minus negative two so magic

implacion

so three plus two that is five x squared

and bring down

the next term and since zero negative

zero x so you're gonna bring down

nothing

negative one okay and then procedurally

taya divide 5x squared divide

x squared the answer is 5 and then

multiply

5 times x squared that is

5 times x squared plus 1 so that is 5x

squared

and 5 times 1 that is positive five

and then subtract okay zero nine

so negative one minus five so apparently

again nothing and negative one minus

five the answer is negative six so

therefore

maritime remainder and negative six so

the final answer is

three x cubed minus two x squared plus

five

minus six over x squared plus one minus

theta okay plus

a negative six in remainder not then so

positive

internation

division x squared plus one okay

so

thank you for watching this video i hope

you learned something

don't forget to like subscribe and hit

the bell button

but updated ko for more video tutorial

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