DIVISION OF POLYNOMIALS USING LONG DIVISION || GRADE 10 MATHEMATICS Q1
Summary
TLDRThis 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
📚 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.
🔢 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.
📘 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.
Mindmap
Keywords
💡Polynomials
💡Long division
💡Dividend
💡Divisor
💡Exponent
💡Partial quotient
💡Remainder
💡Multiplication of polynomials
💡Subtraction in long division
💡Bringing down the next term
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
this is your guide in learning your mod
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