Synthetic Division of Polynomials

Summary

This 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

00:00

📚 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.

05:02

🔍 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.

10:03

🎓 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.

Keywords

💡Synthetic Division

Synthetic division is a method used to divide polynomials, particularly when one of the factors is a linear term like 'x - c'. It is a shortcut for long division and is illustrated in the video through examples. The process involves setting up coefficients of the polynomial and the divisor, performing a series of multiplications and additions, and determining the quotient and remainder. Synthetic division is central to the video's theme of polynomial division.

💡Polynomial Functions

Polynomial functions are algebraic expressions consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents. In the video, polynomial functions are the subject of division using synthetic division, with examples such as 'x^3 - 2x^2 - 5x + 6'.

💡Coefficients

Coefficients are the numerical factors in a polynomial that multiply the variables. In synthetic division, coefficients are crucial as they are the numbers that are manipulated to find the quotient. The video script mentions placing the coefficients of the polynomial inside the division setup, such as '1, -2, -5, 6' for the polynomial 'x^3 - 2x^2 - 5x + 6'.

💡Remainder

In the context of the video, the remainder is the value left over after performing synthetic division. If the remainder is zero, it indicates that the divisor is indeed a factor of the polynomial. For example, when dividing 'x^3 + 5x^2 + 7x + 2' by 'x + 2', a remainder of zero confirms that 'x + 2' is a factor.

💡Divisor

The divisor in the video refers to the expression by which the polynomial is divided, typically a linear term like 'x - c'. The divisor is crucial in synthetic division as it determines the setup of the division process and the potential factorability of the polynomial. For instance, 'x - 3' is the divisor in the first example of the video.

💡Quotient

The quotient is the result of the division process in synthetic division, representing the polynomial that remains after dividing the original polynomial by the divisor. The video demonstrates how to calculate the quotient through a series of multiplications and additions, such as dividing 'x^3 - 2x^2 - 5x + 6' by 'x - 3' to get 'x^2 + x - 2'.

💡Zero of a Function

A zero of a function is a value of the variable that makes the function equal to zero. In the video, finding the zero of the divisor is essential for setting up synthetic division. For example, finding that x = 3 is a zero for 'x - 3' is used to initiate the division process for the polynomial 'x^3 - 2x^2 - 5x + 6'.

💡Descendant Order

In the context of the video, descendant order refers to the arrangement of the polynomial's terms in decreasing powers of the variable. This order is necessary for correctly setting up the synthetic division. The video emphasizes the importance of including all terms, even those with zero coefficients, to ensure the correct division process.

💡Factorable

A polynomial is said to be factorable if it can be expressed as the product of two or more polynomials. The video uses synthetic division to determine if a polynomial is factorable by a given linear term. If the remainder after division is zero, the polynomial is factorable by the divisor, as shown in the examples where 'x + 2' and 'x - 3' are factors.

💡Non-Factorable

Non-factorable, as discussed in the video, refers to a polynomial that cannot be divided evenly by a given divisor using synthetic division, resulting in a non-zero remainder. An example is the division of '3x^2 + 7x - 20' by 'x + 5', which results in a remainder of 20, indicating that the polynomial is not factorable by 'x + 5'.

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