Question

The polynomial kx⁴+ 3x³+ 6 when divided by x - 2 leaves a remainder which is double the
remainder left by the polynomial 2x³ + 17x + k when divided by x - 2. Find the value of k.

Please answer step by step, like how you would explain to a 9th Grade student, cuz that's what I am! I can't understand if you answer confusingly, so please don't beat around the bush.
Thnxx​

Answer 1

Solution :-

Let f(x) = kx⁴+ 3x³+ 6 and g(x) = 2x³ + 17x + k

The question says that when kx⁴+ 3x³+ 6 is divided by x - 2, it leaves a remainder. The remainder obtained here is double the remainder we get by dividing 2x³ + 17x + k by x - 2. So, we have to first divide 2x³ + 17x + k by x - 2.

Finding the zero of (x-2)

x - 2 = 0

x = 2

By remainder theorem -

g(2) = 2 x (2)³ + 17 x 2 + k

      = 2 x 8 + 34 + k

      = 16 + 34 + k

      = 50 + k

The remainder obtained by dividing 2x³ + 17x + k by x - 2 is (50 + k).

Now, dividing kx⁴+ 3x³+ 6 by x - 2.

f(x) = kx⁴+ 3x³+ 6

f(2) = k x (2)⁴+ 3 x (2)³+ 6

     = k x 16 + 3 x 8 + 6

     = 16k + 24 + 6

     = 16k + 30

The remainder obtained by dividing kx⁴+ 3x³+ 6 by x - 2 is (16k + 30).

∴ According To Question -

=> f(2) = 2 x g(2)

Here, f(2) is the remainder obtained by dividing kx⁴+ 3x³+ 6 by x - 2 which is (16k + 30) and g(2) is the remainder obtained by dividing 2x³ + 17x + k by x - 2 which is (50 + k).

In simple terms, f(2) = 16k + 30 and g(2) = 50 + k.

=> 16k + 30 = 2(50 + k)

=> 16k + 30 = 100 + 2k

=> 16k - 2k = 100 - 30

=> 14k = 70

=> k = $\frac{70}{14}$

=> k = 5.

Answer :-

The value of k is 5.