p(X)=2x^3-7x^2+kx+20.the remainder obtained when p(x) is divided by (x-2)and (x-3)are equal?

Question

a)what is k ?
b)what is the remainder when p(x) is divided by(2x-3)?

Answer 1

[07]
By applying remainder theorem,if we substitute x by 2 and 3 respectively,we get the remainder when the expression is divided by x-2 and x-3 respectively
Substituting x by 2 ,we get the remainder
2*8-7*4+2k+20
=16-28+2k+20
=8+2k
Substituting x by 3,we get the remainder
2*27-7*9+3k+20
=54-63+3k+20
=11+3k
Therefore,
11+3k=8+2k
3k-2k=8-11
k= -3
When divided by x-3,the remainder is
11+3k
=11+3(-3)
=11-9=2

Answer 2

p(2)=16-28+2k+20=8+2k which is the reminder for(x-2)
p(3)=54-63+3k+20=11+3k which is the reminder for(x-3)
Because both reminders are equal we have 8+2k=11+3k or k=(-3)
The reminder when p(x) is divided by (2x-3) obtaines if we find p(3/2). p(3/2)=2(3/2)^3-7(3/2)^2-3(3/2)+20=(54/8)-(63/4)-(9/2)+(20)=13/2

Answer 3

a)
A quicker way:
P(3) - P(2) = 2(3^3 - 2^3) - 7(3^2-2^2) + k = 0
=> 38 - 35 + k = 0
=> k = -3

b) P(3/2) = 13/2
------------
Ideas: Use remainder theorem.

Answer 4

try doing a long division!

...... _2x^2_- 3x _+ (k-6) ____________
x-2 | 2x^3 - 7x^2+kx+20
....... 2x^2 - 4x^2, subtract
.......____________
............... -3 x^2 + kx
............... - 3x^2 + 6x, subtract
...............___________
........................... (k-6)x + 20
...........................(k-6)x - 2(k-6), subtract
.......................... ____________
20 + 2(k-6) this is the remainder, or
8 + 2k.

Do this for the other divisor, equate the remainders, to find the value of k.

Once you know this, do another long division by (2x-3).

Answer 5

a) 4
b) 3-4x

Answer 6

the answer is Y

Answer 7

OMG ALIENS HAVE LANDED