Division of Polynomials (Long Division of Polynomials)
Summary
TLDRThe video demonstrates how to solve a polynomial division problem using long division. It walks through dividing the polynomial 2x³ + 3x² - 5x + 6 by x - 2. The process includes dividing the leading terms, multiplying, subtracting, and repeating until reaching the remainder. The final quotient is 2x² + 7x + 9, with a remainder of 24, which is expressed as a fraction over the divisor. The solution is presented as 2x² + 7x + 9 + 24/(x-2).
Takeaways
- 📝 The problem involves solving polynomial division using the long division method.
- 🔢 The dividend is 2x³ + 3x² - 5x + 6, and the divisor is x - 2.
- ➗ The first step is dividing 2x³ by x, resulting in 2x².
- 🔄 Multiplying 2x² by (x - 2) gives 2x³ - 4x², and subtracting it from the dividend results in 7x².
- 🔢 The next step is dividing 7x² by x, which gives 7x.
- 🔄 Multiplying 7x by (x - 2) gives 7x² - 14x, and subtracting it from the current terms results in 9x.
- ➕ Dividing 9x by x results in 9, which is then multiplied by (x - 2), yielding 9x - 18.
- 🔄 Subtracting 9x - 18 from the current terms results in a remainder of 24.
- 🧮 The quotient from the division is 2x² + 7x + 9, and the remainder is 24.
- ✅ The final result is expressed as 2x² + 7x + 9 + 24/(x - 2).
Q & A
What is the given polynomial division problem in the script?
-The problem is to divide the polynomial 2x³ + 3x² - 5x + 6 by the divisor x - 2 using long division.
What is the first step in polynomial long division according to the script?
-The first step is to divide the leading term of the dividend (2x³) by the leading term of the divisor (x), which results in 2x².
How do you multiply the first quotient term by the divisor?
-You multiply 2x² by x - 2, which gives 2x³ - 4x².
What happens after subtracting the first multiplication result?
-After subtracting 2x³ - 4x² from the original terms, you get 0 for the x³ term and 7x² after simplifying the second term.
What is done after bringing down the next term?
-After bringing down -5x, the next step is to divide the new leading term (7x²) by the leading term of the divisor (x), which gives 7x.
What is the result after multiplying the second quotient term by the divisor?
-Multiplying 7x by x - 2 results in 7x² - 14x.
How is the subtraction carried out in the second step of division?
-After subtracting 7x² - 14x from the previous terms, you get 9x by simplifying -5x + 14x.
What is the quotient after the third division?
-Dividing 9x by x results in the next quotient term, which is 9.
How is the final remainder calculated?
-Multiplying 9 by x - 2 results in 9x - 18. After subtracting, the remainder becomes 24.
What is the final answer in polynomial long division?
-The final answer is 2x² + 7x + 9 with a remainder of 24, which can be written as 2x² + 7x + 9 + 24/(x - 2).
Outlines
🔢 Polynomial Long Division Step-by-Step
In this segment, the presenter explains how to solve a polynomial division problem using the long division method. The dividend is the polynomial \(2x^3 + 3x^2 - 5x + 6\) and the divisor is \(x - 2\). The presenter walks through the process of dividing the leading term of the dividend by the leading term of the divisor, which results in \(2x^2\). This is then multiplied by the divisor and subtracted from the original dividend, simplifying the polynomial step-by-step. Each operation is clearly shown, including the handling of signs during subtraction, and the intermediate results are carefully calculated.
✅ Final Calculation and Result Presentation
The presenter continues the long division process, dividing the simplified polynomial \(7x^2 - 5x\) by the divisor's leading term to get \(7x\). This value is multiplied back and subtracted again, resulting in a new polynomial. The process is repeated with the remaining terms, leading to the final quotient \(2x^2 + 7x + 9\) and a remainder of 24. The final answer is expressed as the quotient plus the remainder over the divisor, resulting in the complete solution: \(2x^2 + 7x + 9 + \frac{24}{x-2}\). The presenter highlights the structure of the solution, ensuring clarity in understanding how each component fits into the final answer.
Mindmap
Keywords
💡Long Division of Polynomials
💡Dividend
💡Divisor
💡Quotient
💡Remainder
💡Leading Term
💡Degree
💡Multiplication
💡Subtraction
💡Polynomial
💡Variable
Highlights
Introduction of the long division method for polynomials.
The dividend in the problem is 2x³ + 3x² - 5x + 6.
The divisor in the problem is x - 2.
Start of the long division process by dividing 2x³ by x, resulting in 2x².
Multiplying 2x² by (x - 2), resulting in 2x³ - 4x².
Subtraction of 2x³ - 4x² from the original polynomial, yielding 7x².
Bringing down -5x to get 7x² - 5x.
Dividing 7x² by x to get 7x.
Multiplying 7x by (x - 2), resulting in 7x² - 14x.
Subtracting 7x² - 14x from 7x² - 5x, yielding 9x.
Bringing down the constant term 6, leading to the expression 9x + 6.
Dividing 9x by x to get 9.
Multiplying 9 by (x - 2), resulting in 9x - 18.
Final subtraction yielding the remainder 24.
The final quotient is 2x² + 7x + 9, and the remainder is 24.
The solution is presented in the form of (quotient) + (remainder)/(divisor), which is 2x² + 7x + 9 + 24/(x - 2).
Transcripts
[Musika]
All right so i-solve natin itong problem
na to using the Long Division of
Polynomials so we have here divide p x
so ito Iyung ating Dividend so meron
tayong 2x cu + 3x S - 5x + 6 by d x Ito
naman yung ating
Dividend so meron tayong x - 2 so since
gagamit tayo ng Long Division of
Polynomials gagamitin natin yung long
division
sign ilalagay natin sa loob itong
Dividend na 2 x
cu + 3x
S -
5x + 6 and Dito naman sa labas isusulat
natin ung ating divisor na x -
2 so ang gagawin natin sa process is ang
so 2x cu div x so the answer
is 2x
s and then mmp naman natin itong 2x s
dito sa ating divisor na x - 2 So that's
2x s Tim x so meron tayong
2x cu and then 2x s * -2
meron tayong - 4 x
s so dito sa second line ang
ating operator dito is subtraction or
operation natin dito is subtraction so
ima minus natin ito 2x cu - 2x cu so the
answer is 0 while 3x s
- - 4x s so meron tayong minus sign dito
tapos - 4x s to so magiging positive yan
ano So pwede nating sa isip natin
baliktarin na lang natin yung sign na to
So magiging 3x s + 4x s so we have
7 X
S right and then bring down natin itong
5 x s
so dito magdivide ulit tayo yung first
term or yung leading term nitong ating
7x s- 5x which is 7x idivide natin dito
sa leading term ng ating divisor which
is
x so
7x div x so the answer
is posi
7x so we have +
7x and then mag-multiply ulit tayo 7x *
x so that that's
7 x
s and then 7x Tim
-2 the answer is
-
14
x okay and then tayo ay
magsa-submit x so we have - 5x so
magiging plus
14x so meron tayong
positive
9x right and then bring daw naman natin
itong pos 6 meron na tayong 9x +
6 so magdivide na ulit tayo ang
gagamitin lang ulit natin ay yung
leading term nitong a 9x + 6 div by the
leading term of our
divisor So that's 9x / x the answer is
posi
9 Okay and
then pos 9 or 9 Tim x - 2 dito sa ating
divisor so 9 Tim x so that's 9x
x and then 9 Tim -2 so the answer is
nega
18 Ayan so we have
-1 So ngayon magsa-submit
6
Min
-1 Okay so This is
negative so -
18 so meron ulit tayong dalawang
negative sign so magiging pos 18 siya So
that's 6 + 18 equ to
24 so our quotient
is 2x s
+ 7x + 9 and Our remainder is 24 so we
can write our final answer in the form
of Uh Q of
x or This is our quotient
plus R of x This is our remainder over
the divisor d of
x Okay so our final answer is
2x s + 7x +
9 plus so This is our quotient plus ano
yung remainder
natin
24 div by the divisor which is x-2 so we
write
x-o sa ating
Denominator So this is now our final
answer
so 2x squ + 7x + 9 + 24 over x-2
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