How do I find k if x^4-kx^3-2x^2+x+4 is divided by x-3 and the remainder is 16
2 Answers
hala718 | High School Teacher | (Level 1) Educator Emeritus
Let f(x) = x^4-kx^3-2x^2+x+4
If f(x) is divided by x-3 , then the remainder is 16.
Then we could write:
f(x) = (x-3) * R(x) + 16
Now let us substitute with x =3:
==> f(3) = 0 * R(3) + 16 = 16
==> f(3) = 16 .......(1)
Now let us substitute x=3 in f(x):
f(3) = 3^4 - k* 3^3 -2*3^2 + 3 + 4 = 16
==> 81 - 27k - 18 + 7 = 16
==> -27k = -54
==> k = 2
==> f(x) = x^4 -2x^3 -2x^2 + x + 4
==> f(x) = (x-3)( x^3 + x^2 + x + 4) + 16
neela | High School Teacher | (Level 3) Valedictorian
If x^4-kx^3-2x^2+x+4 is divide by x-3 , we get a remainder 16, then
x^3-kx^3-2x^2+x+4-16 = x^4-kx^3-2x^2+x -12 is perfectly divisible by x-3.
We actually divide :
x-3) x^4-kx^3-2x^2 +x -12( x^3
x^4 -3x^3
----------------------
x-3) (-k+3)x^3 -2x^2( (-k+3)x^2
(-k+3)x^3 - 3(-k+3)x^2
-----------------------------------------------
x-3) (-3k +7)x^2 + x ( ((-3k+7)x
(-3k+7)x^2 -3(-3k+7)x
-----------------------------------------------------------
x-3) (-9k +22)x -12 ( (-9k+22)
(-9k +22)x -3 ( -9k +22)
-----------------------------------------------
0
Therefore -12 +3(-9k+22) = 0
-27k + 54 = 0.
-27k = -54.
k = -54/-27 = 2
or k = 2.
Therefore k = 2 for which x^4-kx^3 -2x^2+x+4 diveided by x-3 gives a remainder of 16.