TAGALOG: Division of Polynomials - Long Division and Synthetic Division #TeacherA
Summary
TLDRIn 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
- đ The lesson focuses on division of polynomials, specifically using long division and synthetic division methods.
- đą The first example demonstrates long division of the polynomial x^2 - 2x - 8 by x - 4, resulting in the quotient x + 2.
- đ 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.
- â The division of the first terms (leading coefficients) is performed, and the result is multiplied by the divisor and subtracted from the dividend.
- đ This process of dividing, multiplying, and subtracting is repeated until the degree of the remaining polynomial is less than the divisor.
- đ Synthetic division is introduced as an alternative method for dividing polynomials, particularly useful for finding specific roots.
- đ For synthetic division, coefficients of the dividend are aligned, and the divisor's root is used to successively calculate new coefficients and the remainder.
- đ 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.
- đ 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.
- đšâđ« 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
đ 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.
đ 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.
đ 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.
đ 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.
Mindmap
Keywords
đĄPolynomials
đĄLong Division Method
đĄSynthetic Division
đĄDividend
đĄDivisor
đĄQuotient
đĄRemainder
đĄCoefficients
đĄExponents
đĄVariables
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
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