Just a quick question about a questoin relating to factor theorem:

Find the value of k so that when http://img250.imageshack.us/img250/6594/untitledfb5.jpg is divided by x + k, the remainder is 3.

Thanks! :)

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- Feb 13th 2007, 04:54 PMty2391Factor Theroem
Just a quick question about a questoin relating to factor theorem:

Find the value of k so that when http://img250.imageshack.us/img250/6594/untitledfb5.jpg is divided by x + k, the remainder is 3.

Thanks! :) - Feb 13th 2007, 05:10 PMThePerfectHacker
The division algorithm says we can find the quotient and remainder:

x^3+5x^2+6x+11 = q(x)(x+k)+3

Since these are equal, their evaluations are equal as well.

Let us evaluate this at x=-k.

Thus,

(-k)^3+5(-k)^2+6(-k)+11=q(-k)(0)+3

-k^3+5k-6k+11=3

k^3-5k+6k-8=0

Use the rational root theorem and see that k=4 is a solution. (In fact the only one).

Thus, k=4. - Feb 13th 2007, 08:37 PMSoroban
Hello, ty2391!

Quote:

Find the value of*k*so that when x³ + 5x² + 6x + 11 is divided by x + k, the remainder is 3.

. . If f(a) = r, then f(x) ÷ (x - a) has a remainder of*r*.

We want to find*a*so that f(a) = 3.

If*a*is nonpositive, we can see that f(a)__>__11.

. . Hence,*a*must be negative.

We find that: . f(-1) .= .9

. . . . . . . . . . .f(-2) .= .11

. . . . . . . . . . .f(-3) .= .13

. . . . . . . . . . .f(-4) .= . 3 . ← There!

Therefore: .**k = 4**

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Check

(x³ + 5x² + 6x + 11) ÷ (x + 4) . = . x² + x + 2, .remainder 3