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# 6. Consider the cubic polynomial p(x) = x^3 + x^2 - 46x + 80.

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6. Consider the cubic polynomial p(x) = x^3 + x^2 - 46x + 80.

(a) Using polynomial long division, write the ratio of p(x)/(x-3) in quotient-remainder form, i.e. in the form q(x) + (r)/(x-3). Evaluate p(3). How does this help you check your quotient-remainder form?

Feb 10, 2019

#1
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x^2+4x-34       -  22/(x-3)

x-3  |  x^3+x^2-46x+80

x^3 -3x^2

_______________

4x^2-46x

4x^2-12x

_____________

-34x+80

-34x+102

__________

(remainder)  -22

Basically with remainder  -22/(x-3)     then  p(3)  will equal -22

See cphill's explanation/answer here for the second part:

https://web2.0calc.com/questions/what-does-this-mean_10

Feb 10, 2019
edited by Guest  Feb 10, 2019
edited by ElectricPavlov  Feb 10, 2019
#2
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Using EP's answer of R = -22

We should get the same answer by subbing 3 inito the polynomial....so...

(3)^3 + (3)^2 - 46(3) + 80 =

27 + 9 -  138 + 80 =

116 - 138 =

-22       ......Magic !!!!!   Feb 10, 2019
#3
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Wait... I don't understand by the question: How does this help you check your quotient-remainder form?

Feb 10, 2019
#4
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It helps because     the    "r"  in   r / [ x - 3 ]     should be the same as evaluating P(a)

So...in the case EP divided the polynomial by x - a  =  x - 3        got  a remainder of -22

Then....we can check whether this remainder is correct by evaluating P(a) = P(3)

Note that when we evaluated P(3)....we put 3 into the polynomial and got -22

So....we can be sure that this remainder is correct....!!!!!

Does that make sense, GM???   CPhill  Feb 10, 2019
edited by CPhill  Feb 10, 2019
#5
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Ahh...Everything makes perfect sense now! Thanks a lot!