Synthetic Division of Polynomials
Summary
TLDRThis educational video tutorial teaches viewers how to use synthetic division to divide polynomial functions. It guides through step-by-step examples, starting with dividing \(x^3 - 2x^2 - 5x + 6\) by \(x - 3\), and progresses to more complex polynomials. Each example illustrates the process of setting up coefficients, performing the division, and interpreting the remainder to determine if the polynomial is factorable. The video also highlights common mistakes, such as forgetting to include zero coefficients, and emphasizes the importance of order in polynomial functions. By the end, viewers should have a clear understanding of synthetic division and its application in polynomial division.
Takeaways
- 📚 Synthetic division is a method used to divide polynomial functions.
- 🔢 Start by writing down the coefficients of the dividend polynomial in descending order.
- 🎯 Identify the zero of the divisor (x - c) by setting it equal to zero and solving for x.
- ✏️ Place the zero found (c) at the top of the division process and bring down the first coefficient.
- 🔄 Perform the synthetic division steps: multiply, add, bring down, repeat until the end.
- 🧮 If the remainder is zero, the divisor is a factor of the polynomial; if not, it's not a factor but division still occurs.
- 📉 The quotient from the division is the resulting polynomial after the division process.
- 🔄 For non-zero remainders, include the remainder divided by the divisor as part of the final quotient.
- ⚠️ Remember to include all coefficients, including zero coefficients, when setting up the division.
- 📈 Practice with various examples to understand how synthetic division works with different polynomials and divisors.
Q & A
What is the main focus of the video?
-The main focus of the video is to teach how to divide polynomial functions using synthetic division.
What is the first polynomial function given in the video example?
-The first polynomial function given in the video example is x^3 - 2x^2 - 5x + 6.
By what factor is the first polynomial function divided in the video?
-The first polynomial function is divided by x - 3.
What does the number 3 represent in the synthetic division process for the first example?
-The number 3 represents the value of x that makes the divisor x - 3 equal to zero, which is one of the zeros of the function.
What is the result of the synthetic division for the first example?
-The result of the synthetic division for the first example is x^2 + x - 2.
What does a remainder of zero in synthetic division indicate?
-A remainder of zero in synthetic division indicates that the divisor is a factor of the polynomial, meaning the polynomial is factorable by the divisor.
What is the second polynomial function discussed in the video?
-The second polynomial function discussed is x^3 + 5x^2 + 7x + 2.
What is the divisor used for the second polynomial function in the video?
-The divisor used for the second polynomial function is x + 2.
What is the result of the synthetic division for the second example?
-The result of the synthetic division for the second example is x^2 + 3x + 1.
How is the remainder handled in synthetic division if it's not zero?
-If the remainder is not zero, it is added to the result as a fraction with the divisor in the denominator.
What is the importance of writing the polynomial function in descending order when performing synthetic division?
-Writing the polynomial function in descending order ensures that the correct coefficients are used in the synthetic division process, which is crucial for obtaining the correct result.
What is the final step in the synthetic division process after obtaining the quotient?
-The final step in the synthetic division process after obtaining the quotient is to add the remainder (if not zero) as a fraction with the divisor in the denominator to complete the division.
Outlines
📚 Introduction to Synthetic Division
This paragraph introduces the concept of synthetic division, a method used to divide polynomial functions. The video provides a step-by-step guide on how to perform synthetic division using an example: dividing the polynomial x^3 - 2x^2 - 5x + 6 by x - 3. The process involves writing down the coefficients of the polynomial, bringing down the leading coefficient, and performing a series of multiplications and additions. The example concludes by demonstrating how a remainder of zero indicates that the divisor is a factor of the polynomial, resulting in a quotient of x^2 + x - 2.
🔍 Further Examples of Synthetic Division
The second paragraph presents additional examples of synthetic division, reinforcing the method with different polynomials. It includes dividing x^3 + 5x^2 + 7x + 2 by x + 2, demonstrating that the polynomial is factorable by x + 2, leading to a quotient of x^2 + 3x + 1. The paragraph also covers a case where the polynomial 3x^2 + 7x - 20 is not factorable by x + 5, resulting in a quotient of 3x - 8 with a remainder of 20. The process of handling the remainder is explained, showing how to express the result as a combination of the quotient and the remainder divided by the divisor.
🎓 Advanced Synthetic Division Techniques
The third paragraph tackles more complex synthetic division problems, emphasizing the importance of including zero coefficients when the polynomial's degree is higher than the divisor's degree. It guides through the division of 7x^3 + 6x - 8 by x - 4, highlighting the correct order of coefficients, including the zero terms. The process results in a quotient of 7x^2 + 28x + 118 and a remainder of 464, which is then expressed as part of the final answer. Another example with 3x^4 - 5x^2 + 6 divided by x - 2 is also provided, showing the inclusion of higher degree zero coefficients and resulting in a quotient of 3x^3 + 6x^2 + 7x + 14 with a remainder of 34.
🏁 Conclusion of Synthetic Division Tutorial
The final paragraph wraps up the tutorial on synthetic division, summarizing the key points and techniques covered in the video. It reiterates the importance of synthetic division as a method for dividing polynomials and determining factors. The paragraph concludes by thanking viewers for watching and encouraging them to apply the learned techniques in their mathematical endeavors.
Mindmap
Keywords
💡Synthetic Division
💡Polynomial Functions
💡Coefficients
💡Remainder
💡Divisor
💡Quotient
💡Zero of a Function
💡Descendant Order
💡Factorable
💡Non-Factorable
Highlights
Introduction to dividing polynomial functions using synthetic division.
Example of dividing x^3 - 2x^2 - 5x + 6 by x - 3 using synthetic division.
Explanation of placing coefficients and the divisor's root in synthetic division.
Step-by-step process of synthetic division for the given example.
Conclusion that x - 3 is a factor if the remainder is zero after synthetic division.
Result of the division: x^2 + x - 2.
Second example with polynomial x^3 + 5x^2 + 7x + 2 and divisor x + 2.
Demonstration of synthetic division for the second example.
Verification that x + 2 is a factor with a zero remainder.
Division result: x^2 + 3x + 1.
Third example with polynomial 3x^2 + 7x - 20 and divisor x + 5.
Synthetic division process for the third polynomial.
Explanation of non-zero remainder and its implications for factorability.
Division result with a remainder: 3x - 8 + 20/(x + 5).
Fourth example with polynomial 7x^3 + 6x - 8 and divisor x - 4.
Emphasis on including zero terms in polynomial for synthetic division.
Synthetic division for the fourth example, including handling zero terms.
Final division result: 7x^2 + 28x + 118 + 464/(x - 4).
Fifth and final example with polynomial 3x^4 - 5x^2 + 6 and divisor x - 2.
Synthetic division for the fifth example, including handling multiple zero terms.
Final division result: 3x^3 + 6x^2 + 7x + 14 + 34/(x - 2).
Conclusion of the video with a summary of synthetic division.
Transcripts
in this video we're going to focus on
dividing polynomial functions
using synthetic division
so let's start with this example
let's say we have x cubed
minus 2x squared
minus five x
plus six
and let's divide it
by
x minus three
using synthetic division
how can we do this
well first
let's draw this
on the inside you want to put the
coefficients
of
this function
so it's 1x cubed minus two x squared
minus five x plus six
now you need to put a number here
what number do you think goes in that
region
if you take this factor x minus three
and if you set it equal to 0 and solve
for x
you'll see that x is equal to 3.
that's one of the zeros of the function
at least sometimes it's a zero
but basically if you see minus 3 reverse
it
so in this case we're gonna use positive
three
so let's bring down a one
three times one is three
negative two plus three is one
and then multiply 3 times 1 is 3
and then add so you're going to multiply
add multiply add and keep doing that
negative 5 plus 3 is
negative 2 and then multiply
3 times negative 2 is negative 6.
and a remainder is zero if the remainder
is zero that means that this is a factor
of this function
which means three is one of the zeros
if you don't get a zero here then this
is not a factor
it's not factorable by x minus three but
if you do get a zero
this function is factorable by x minus
three so something to keep in mind
so what does this all mean
now x cubed divided by x
is x squared
so the first number is the coefficient
for x squared so we have 1x squared
and then
plus 1x with a constant of negative two
and so this
is the answer
x cubed minus two x squared minus five x
plus six
divided by x minus three
is equal to x squared plus x minus two
now let's try another example
let's take x cubed plus five x squared
plus seven x
plus two
and let's divide it by
x plus two
so for the sake of practice go ahead and
try it
feel free to pause the video
so if you set x plus two equal to zero
and solve for x you'll get negative two
so let's put that number out in the
front
and the coefficients for this
polynomial expression is going to be one
five
seven and two
so let's bring down the one
and let's multiply negative two times
one
is negative two and then add five plus
negative two is three and then multiply
negative two times three
is negative six
and seven plus negative six is one
and then negative two times one is
negative two
and two and negative two cancels so once
again we have a remainder of zero
which means
this expression is factorable by x plus
two
so the answer that we're looking for
it's one x squared
plus
three x plus one
so this is the solution
after you divide by these two
expressions
now let's move on to our third example
three x squared plus seven x
minus
twenty divided by x plus five
go ahead and divide these two using
synthetic division
so first let's set x plus five equal to
zero
so if we subtract both sides by five
we're going to use negative five
and the coefficients are three seven
and negative twenty so let's bring down
the three first
negative five times three
that's negative 15.
and then 7 plus negative 15
is negative 8
and then negative 5 times negative 8
that's positive 40.
and negative 20 plus 40 that's 20. so
this time
the remainder is not zero it's 20.
so this expression is not factorable by
x plus five
nevertheless we can still divide it
so after we divide these two functions
or these two expressions what do we get
3x squared divided by x is 3x
so this is not going to be 3x squared
this is going to be 3x and then minus 8.
now
if you have a remainder that's not 0
what do you do with it
it's going to be the remainder
20 divided by what you try to divide it
by which is x plus 5.
so
3x squared plus 7x minus 20 divided by x
plus 5
is equal to this expression
3x minus 8
plus
20 divided by x plus 5.
so that's what you need to do if you
don't have a remainder of zero
need to add r
divided by whatever you try to divide it
by
so this is the answer for the problem
number four seven x cubed
plus six x
minus eight
divided by x minus four
now this one is a little different than
the last three problems
so go ahead and try this problem but be
careful
so once again if we set x minus four
equal to zero
if we add four to both sides x will
equal four
so let's put a four on the outside
now what are the coefficients that we
need to write for this problem
now if you put 7 6 and negative 8 you
won't get the answer right
when you write the polynomial function
you need to be aware of zero x squared
so you need to write this in descendant
order the zero is very important
so it's going to be seven
zero
six negative eight
so if you forget this zero your answer
will be different than what it should be
so let's bring down the seven
four times seven is twenty eight
and zero plus twenty eight is twenty
eight
now four times twenty eight
four times twenty is eighty four times
eight is thirty two
eighty plus thirty two that's going to
be one twelfth
and six plus one twelve is one eighteen
now four times one eighteen
so four times a hundred is four hundred
four times ten is forty and four times
eight is thirty two
so forty and thirty two is seventy two
plus four hundred that's going to be
four seventy two
and negative eight plus four seventy two
is 464.
and so that is the remainder
now 7x cubed divided by x is 7x squared
so that's going to give us the first
term
it's going to be 7x squared
and then plus 28x
plus a constant of 118
plus the remainder of 464
divided by what you try to divide by x
minus 4.
so this is the final answer of the
problem
now let's work on one more example
three x to the fourth
minus five x squared
plus six
divided by x minus two
it's very similar to the last example so
for the sake of practice pause the video
and try this problem
so let's set x minus two equal to zero
so adding two to both sides x is two
now keep in mind
don't forget about the zero x cubed
and also
zero x
so there's two zeros that we have to
deal with
in this synthetic division problem so we
have three
zero
negative five zero and six
so let's bring down the three
two times three is six
zero plus six is six
and then two times six is twelve
negative five plus twelve is seven
two times seven is fourteen
zero plus fourteen is fourteen
and then two times fourteen is twenty
eight
6 plus 28 is 34.
now
3x to the fourth divided by x is going
to be 3x cubed
so let's bring this down this is going
to be 3x cubed and in descending order
the next one is going to be 6x squared
plus 7x
plus 14
and we have a remainder
of 34.
so it's going to be plus the remainder
divided by what you try to divide it by
x minus two
and so this is the final answer so three
x to the fourth minus five x squared
plus six
divided by x minus two is equal to
everything that you see in the blue box
and that's it for this video so now you
know how to divide using synthetic
division hopefully you found it to be
helpful so thanks for watching
you
Browse More Related Video
Polinomial (Bagian 4) - Teorema Sisa dan Teorema Faktor
Pembagian suku banyak dengan cara bersusun - Menentukan hasil dan sisa pembagian
Division of Polynomials (Long Division of Polynomials)
How to Divide Polynomials Using LONG DIVISION | Math 10
TAGALOG: Division of Polynomials - Long Division and Synthetic Division #TeacherA
How to Divide Polynomials using Long Division - Polynomials
5.0 / 5 (0 votes)