I need to know how to write a cubic function with a remainder of 8 for f(2) and a remainder of -5 for f(3) using synthetic division.

Expert Answers
embizze eNotes educator| Certified Educator

We are asked to find a cubic function such that f(2) has remainder 8 and f(3) has remainder -5.

Let the cubic be x^3+ax^2+bx+c. (There are an infinite number of cubics through 2 points -- we will choose one with leading coefficient 1.)

Using synthetic division:

2 | 1       a        b                 c
   ------------------------------
     1    (a+2)  2a+b+4  4a+2b+c+8

So we know 4a+2b+c=0. Now try 3:

3 | 1       a           b               c
   ------------------------------------
     1     a+3    3a+b+9    9a+3b+c+27

Then 9a+3b+c=-32

We have 2 equations in 3 unknowns -- solving the system we find that there are an infinite number of possible solutions (as expected) of the form:

(a,-5a-32,6a+64)

Let a=1; then b=-37 and c=70 to get a cubic:

`f(x)=x^3+x^2-37x+70 `

Note that f(2)=8 and f(3)=-5 as required.

Let a=3; then b=-47 and c=82.

This cubic is `f(x)=x^3+3x^2-47x+82 `

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One solution to the problem is `f(x)=x^3+x^2-37x+70 `

We can characterize all solutions with leading coefficient 1:

`f(x)=x^3+ax^2+bx+c ` where a is any real number, b=-5a-32 and c=6a+64. There are other answers where the leading coefficient is not 1.

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embizze eNotes educator| Certified Educator

A note to iamkaori: Your set up is correct, but in synthetic division you add, not subtract. You can check your answer to see that it does not meet the given conditions.

iamkaori | Student

Work your way backwards.

For a cubic function with a remainder of 8 for f(2), we know that:

2) A  B   C   D

+      2A  2E 2F

-------------------

    A  E    F   8

and that Ax^3+Bx^2+Cx+D will be the equation.

Now plug in any number in F, and determine what D would be with D+2F being 8.

For example, if you plug 10 into F, it would be:

2) A  B   C    -12

+      2A  2E  20

-------------------

    A  E    10   8

now plug another number into E, for instance 9:

2) A  B    -8    -12

+      2A  18   20

-------------------

    A  9   10   8

plug another number into A, for instance 4:

2) 1  1    -8    -12

+      8   18   20

-------------------

    1  9   10   8

Now, the equation would be:

x^3+x^2-8x-12


Hope this helped, now you can try the latter yourself!

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