When p(x)=2x^4 + kx +1 is divided by x+1, the remainder is the same as when q(x)=3x^4 + kx2 +2 is divided by x–1. Find k.

Answer 1

Using synthetic division the remainder of p(x) is -k + 3

Using synthetic division the remainder of q(x) is k + 5

-k + 3 = k + 5

Can you solve for k and answer?

Answer 2

I need to find the value of k using the Remainder Theorem.

Remainder Theorem: When a polynomial f(x) is divided by (x - a), the remainder is f(a).

For p(x) = 2x⁴ + kx + 1 divided by (x + 1):

Dividing by (x + 1) means dividing by (x - (-1)), so I evaluate p(-1):

p(-1) = 2(-1)⁴ + k(-1) + 1 p(-1) = 2(1) - k + 1 p(-1) = 3 - k

For q(x) = 3x⁴ + kx² + 2 divided by (x - 1):

I evaluate q(1):

q(1) = 3(1)⁴ + k(1)² + 2 q(1) = 3 + k + 2 q(1) = 5 + k

Setting the remainders equal:

Since the remainders are the same:

3 - k = 5 + k

3 - 5 = k + k

-2 = 2k

k = -1

Verification:

1. p(-1) = 3 - (-1) = 4
2. q(1) = 5 + (-1) = 4 ✓

Therefore, k = -1.