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.

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embizze's profile pic

embizze | High School Teacher | (Level 1) Educator Emeritus

Posted on

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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iamkaori's profile picture

iamkaori | Student, Grade 9 | (Level 2) Salutatorian

Posted on

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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embizze's profile pic

embizze | High School Teacher | (Level 1) Educator Emeritus

Posted on

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.

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