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Using synthetic division to divide a second-degree polynomial

Use synthetic division to divide 5 x 2 3 x 36 by x 3.

Begin by setting up the synthetic division. Write k and the coefficients.

A collapsed version of the previous synthetic division.

Bring down the lead coefficient. Multiply the lead coefficient by k .

The set-up of the synthetic division for the polynomial 5x^2-3x-36 by x-3, which renders {5, -3, -36} by 3.

Continue by adding the numbers in the second column. Multiply the resulting number by k . Write the result in the next column. Then add the numbers in the third column.

Multiplied by the lead coefficient, 5, in the second column, and the lead coefficient is brought down to the second row.

The result is 5 x + 12. The remainder is 0. So x 3 is a factor of the original polynomial.

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Using synthetic division to divide a third-degree polynomial

Use synthetic division to divide 4 x 3 + 10 x 2 6 x 20 by x + 2.

The binomial divisor is x + 2 so k = −2. Add each column, multiply the result by –2, and repeat until the last column is reached.

Synthetic division of 4x^3+10x^2-6x-20 divided by x+2.

The result is 4 x 2 + 2 x 10. The remainder is 0. Thus, x + 2 is a factor of 4 x 3 + 10 x 2 6 x 20.

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Using synthetic division to divide a fourth-degree polynomial

Use synthetic division to divide 9 x 4 + 10 x 3 + 7 x 2 6 by x 1.

Notice there is no x -term. We will use a zero as the coefficient for that term.

..

The result is 9 x 3 + x 2 + 8 x + 8 + 2 x 1 .

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Use synthetic division to divide 3 x 4 + 18 x 3 3 x + 40 by x + 7.

3 x 3 3 x 2 + 21 x 150 + 1 , 090 x + 7

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Using polynomial division to solve application problems

Polynomial division can be used to solve a variety of application problems involving expressions for area and volume. We looked at an application at the beginning of this section. Now we will solve that problem in the following example.

Using polynomial division in an application problem

The volume of a rectangular solid is given by the polynomial 3 x 4 3 x 3 33 x 2 + 54 x . The length of the solid is given by 3 x and the width is given by x 2. Find the height, t , of the solid.

There are a few ways to approach this problem. We need to divide the expression for the volume of the solid by the expressions for the length and width. Let us create a sketch as in [link] .

Graph of f(x)=4x^3+10x^2-6x-20 with a close up on x+2.

We can now write an equation by substituting the known values into the formula for the volume of a rectangular solid.

V = l w h 3 x 4 3 x 3 33 x 2 + 54 x = 3 x ( x 2 ) h

To solve for h , first divide both sides by 3 x .

3 x ( x 2 ) h 3 x = 3 x 4 3 x 3 33 x 2 + 54 x 3 x ( x 2 ) h = x 3 x 2 11 x + 18

Now solve for h using synthetic division.

h = x 3 x 2 11 x + 18 x 2

The quotient is x 2 + x 9 and the remainder is 0. The height of the solid is x 2 + x 9.

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The area of a rectangle is given by 3 x 3 + 14 x 2 23 x + 6. The width of the rectangle is given by x + 6. Find an expression for the length of the rectangle.

3 x 2 4 x + 1

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Key equations

Division Algorithm f ( x ) = d ( x ) q ( x ) + r ( x )  where  q ( x ) 0

Key concepts

  • Polynomial long division can be used to divide a polynomial by any polynomial with equal or lower degree. See [link] and [link] .
  • The Division Algorithm tells us that a polynomial dividend can be written as the product of the divisor and the quotient added to the remainder.
  • Synthetic division is a shortcut that can be used to divide a polynomial by a binomial in the form x k . See [link] , [link] , and [link] .
  • Polynomial division can be used to solve application problems, including area and volume. See [link] .

Questions & Answers

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Sanjika
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Sanjika
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SharaAmor
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Tina
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Usman
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First and foremost -- Wow Rowdy! You have been through a highly difficult situation in your life, hands down. As a wife, I simply cannot phathom nor imagine slightly the feelings and thoughts you bare and I'm sure as these words do not help and hit rewind; my heart and prayers are going out to you-
Stephanie
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Stephanie
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Stephanie
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Stephanie
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Ali Reply
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Ali
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Ali
Information about child psychology
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Geo
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Jobin
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QSN Reply
neuropsychology
jeddy
behaviorism
Mira
Legal psychology
Miljana
Developmental psychology, Sports psychology , Clinical psychology , Marketing psychology , counseling psychology , Biological psychology, Educational psychology , Positive psychology etc.
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child.psychology
Nss
social, occupational psychology
jeddy
what does it mean when the temporal lobe is anterior to occipital lobe
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the cortex refers to what part of the brain
John
the cortex refers to what part of the brain
John
cerebral cortex
jeddy
behind?
jeddy
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marjelee Reply
during bad experience
Nyakuei
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gokul
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rhoda
I would think that training should be an ungoing thing. That's before, during and after. just in case there's a reoccurrence
Angela
any time
Aaronz
what is Karma
ana Reply
I think the summation of good and bad deeds you do in your life. Accidentally or otherwise.
Swarada
Yes. I totally agree.
ana
And if it is bad then the root must have been too.
ana
Yes , In Hinduism we also believe that our parent's Karma also applies to us ! Don't know about that , but it kept people in check .
Swarada
Adding to Swarada point. karma can be good or bad. both will have consequences. karma is biasedness you have shown in your life. like a king could not have two rules for normal man and his son, that comes under karma.
Sujeet
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Swarada
Ok. It is also said that when one does something... bad or good..it will come back to you 10 folds of the same.
ana
yea that too ! I remember my mother always said if u stole someone's 10 rupees ur 100 will be stolen ( excuse my English ! I'm a non - English speaking person)
Swarada
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Alisha
certainly
Crazy
more important.. this concept was told by lord Krishna as part of spirituality, to focus on your good deed ,the purpose you are born for and your responsibilities. Not for hoping that bad will happen with people who made you feel bad. just giving example ;)
Sujeet
karma is the boomerang of your actions. what u put out in the universe u get back
David
Who is the founder of psychology
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Wilhelm wundt
Amanda
Wilhelm Wundt
Sneha
William wundth
Ihrar
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and William James
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Wilhelm Wundt
adam
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Yes and also the "Father of Psychology" I think the correct word is Father not Founder.
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oh ok. noted
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i believe it goes beyond him I believe it is Ancient but un formalized
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I like your answer Swarada
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Oh thanks! But u summarised it well!
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what is up my fellow psych peeps!!
what are some academic research on developmental psychology
Chezy
Sawrada Founder of Psychology is W.Wundt but Father of Psychology is Sigmund Freud...
Wajid
what are some good schools to study psychology?
Enjoy Reply
George Mason, Penn State, UVA, university of Michigan
Jasmine
Hi I'm currently doing my experimental psychology research can you suggest a topic or research topic for my research thank you and please see 😊
Jerico Reply
Hello Jerico. I could help you out with some ideas if you're interested!
Yasmine
can u give me some ideas too?
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For sure. It depends if you're trying to do an experiment (using statistics probably) or a case study what are you aiming for?
Yasmine
Hello Jerico. Try to opt for something simple and easy to do. I wouls say perhaps study the effects of color on concentration? You could get a group of people and have them take a memory test (you can find some online) in a white room for example. Then have them take another memory test
Yasmine
similar to the first one but in a different room and then you check the responses if the performance is better
Yasmine
but you have to know that a lot of factors will interfere in the performance such as mental state etc so make sure to choose a certain specific population. that's very important
Yasmine
you could be a bit more specific about it , to isolate the variables that can mess it up like age, sex, etc... you gotta take that into account
Yasmine
so that the results get somewhat accurate
Yasmine
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Jerico
You're very welcome
Yasmine
what are your interests in general?
Yasmine
Althogh I'm persuing botany and engineering.... but I also want to study human behaviour and psychology.... I would like if anyone can share their knowledge with me !!!!
Crazy
I'm interested on effect of electroconvulsive therapy on people having a depression
Jerico
What is Electroconvulsive therapy?
Pixeled
Astral Projection
Mansi
Electroconvulsive therapy (ECT) is a medical treatment that is most commonly used in patients with severe major depression.
Jerico
What are the expected dutties functions and responsibilities of a head of an agency based on the principles, Practice of industrial psychology?
Paul Reply
what are the psychological characteristics expected from an employee wether regular or occupying a position?
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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