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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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ology means a study of
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a branch of knowledge
psych means soul and ology means to study.
Can someone help me with tha latent functions of different social institutions
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Interestingly Reply
Factors that affect learning?
Ofentse Reply
I am struggling with survivors guilt and complicated grief after traumatic death of spouse. how to cope.
Rowdy Reply
Try to talk with someone you trust and join people who makes you feel worthy. try to help someone. Helping someone makes you feel better
I think you must seek professional help.
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stop burdening yourself with thoughts that say you could have done something..you couldn't..your just a human being and you cant always be a superman who saves the day..please dont feel guilty when you have no reason to..(cont.)
learn to love yourself again..learn to live ..for yourself,for the people who love you and need you in their lives.live for people who you once loved...Think about how you can make your life meaningful once again.It might seem impossible but trust me its necessary ..
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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-
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to guide you through; keep an your senses wide open so that you can learn when she is trying to communicate.
What is meant by politics of intelligence? Please guide about this article.
Ali Reply
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Can u please explain it to me in your words?
Information about child psychology
Mohammed Reply
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QSN Reply
Legal psychology
Developmental psychology, Sports psychology , Clinical psychology , Marketing psychology , counseling psychology , Biological psychology, Educational psychology , Positive psychology etc.
social, occupational psychology
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John Reply
the cortex refers to what part of the brain
the cortex refers to what part of the brain
cerebral cortex
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marjelee Reply
during bad experience
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training is possibly warranted for all situations.
I would think that training should be an ungoing thing. That's before, during and after. just in case there's a reoccurrence
any time
what is Karma
ana Reply
I think the summation of good and bad deeds you do in your life. Accidentally or otherwise.
Yes. I totally agree.
And if it is bad then the root must have been too.
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 .
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.
Oh perfectly said!
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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)
karma is nothing but what one usually mean when they say- what you sow, so shall you reap.
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 ;)
karma is the boomerang of your actions. what u put out in the universe u get back
Who is the founder of psychology
Daryl Reply
Wilhelm wundt
Wilhelm Wundt
William wundth
Wilhelm Wundt
and William James
Wilhelm Wundt
Wilhelm Wundt is the founder of structuralism
Yes and also the "Father of Psychology" I think the correct word is Father not Founder.
oh ok. noted
i believe it goes beyond him I believe it is Ancient but un formalized
I like your answer Swarada
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what are some academic research on developmental psychology
Sawrada Founder of Psychology is W.Wundt but Father of Psychology is Sigmund Freud...
what are some good schools to study psychology?
Enjoy Reply
George Mason, Penn State, UVA, university of Michigan
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
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can u give me some ideas too?
For sure. It depends if you're trying to do an experiment (using statistics probably) or a case study what are you aiming for?
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
similar to the first one but in a different room and then you check the responses if the performance is better
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
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
so that the results get somewhat accurate
that is very detailed information I really like it thank you very much 😊 your so very kind
You're very welcome
what are your interests in general?
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 !!!!
I'm interested on effect of electroconvulsive therapy on people having a depression
What is Electroconvulsive therapy?
Astral Projection
Electroconvulsive therapy (ECT) is a medical treatment that is most commonly used in patients with severe major depression.
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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