the polynomials P(x)= kx^3+3x^2-3 and Q(x)= 2x^3-5x+k when divided by (x-4) leave the same remainder the value of k is
p(4) = 64k + 48 - 3
= 64k + 45
Q(4) = 128 - 20 + k
= k + 108
by the remainder theorem , each of these are the remainders when divided by x-4
But they are equal ...
64k + 45 = k + 108
63k = 63
k = 1
To find the value of k, we need to equate the remainders when dividing the polynomials P(x) and Q(x) by (x-4).
Let's start by dividing P(x) by (x-4):
P(x) = kx^3 + 3x^2 - 3
Dividing by (x-4), we use synthetic division:
4 │ k 3 0 -3
│ 4k 28k 112k-460
───────────────────────
k 7k 28k+112k-463
The remainder is given by 28k + 112k - 463.
Now let's divide Q(x) by (x-4):
Q(x) = 2x^3 - 5x + k
Dividing by (x-4) using synthetic division:
4 │ 2 0 -5 k
│ 8 32 108+4k
─────────────────────
2 8 27+4k 112+4k
The remainder is given by 27 + 4k.
Since P(x) and Q(x) leave the same remainder when divided by (x-4), we have:
28k + 112k - 463 = 27 + 4k
Combining like terms:
140k - 436 = 27 + 4k
Subtracting 4k from both sides:
136k - 436 = 27
Adding 436 to both sides:
136k = 463
Dividing both sides by 136:
k = 463/136
Therefore, the value of k is approximately 3.408.
To find the value of k, we need to set up an equation based on the given information.
When a polynomial is divided by (x - a), the remainder is given by substituting x = a into the polynomial and evaluating it. So, we need to find the remainder when both P(x) and Q(x) are divided by (x - 4) and set them equal to each other.
Let's first find the remainder when P(x) is divided by (x - 4). Substitute x = 4 into P(x):
P(4) = k(4^3) + 3(4^2) - 3
= 64k + 48 - 3
= 64k + 45
Now, let's find the remainder when Q(x) is divided by (x - 4). Substitute x = 4 into Q(x):
Q(4) = 2(4^3) - 5(4) + k
= 2(64) - 20 + k
= 128 - 20 + k
= 108 + k
Since both P(x) and Q(x) leave the same remainder when divided by (x - 4), we can set their remainders equal to each other:
64k + 45 = 108 + k
Now, we can solve this equation for k:
64k - k = 108 - 45
63k = 63
k = 1
Therefore, the value of k is 1.