Question

Find the value of k, if \[{x^2} + 2x = 3k\] has a remainder 3 when divided by x – 4.
(a). 5
(b). 6
(c). 7
(d). 8

Answer 1

- Hint: Substitute x = 4 in the given expression and find the resulting expression. Equate the resulting expression to the given number 3 and solve to find the value of k.

Complete step-by-step solution -

It is given that when the equation \[{x^2} + 2x = 3k\] is divided by the polynomial x – 4, the value of the remainder is 3.
Let us find the polynomial which is divided by x – 4 from \[{x^2} + 2x = 3k\] by taking all terms to the left-hand side of the equation.
\[{x^2} + 2x - 3k = 0\]
Hence, the polynomial is \[{x^2} + 2x - 3k\]. This is the polynomial which when divided by x – 4 gives the remainder 3.
To find the value of k, we can substitute the value of x as 4 in the polynomial \[{x^2} + 2x - 3k\] and the final result should be equal to the value of the remainder because x – 4 exactly divides the polynomial \[{x^2} + 2x - 3k - 3\] and x – 4 is the zero of the given polynomial minus 3. Hence, we have:
\[{(4)^2} + 2(4) - 3k = 3\]
Simplifying the above equation, we get:
\[16 + 8 - 3k = 3\]
\[24 - 3k = 3\]
Taking 3k to the right-hand side and 3 to the left-hand side, we get:
\[3k = 24 - 3\]
\[3k = 21\]
Dividing 21 by 3, we get the value of k as follows:
\[k = \dfrac{{21}}{3}\]
\[k = 7\]
The value of k is 7.
Hence, the correct answer is option (c).

Note: You can also divide the polynomial \[{x^2} + 2x - 3k\] by x – 4 using long division and then find the remainder in terms of k and then equate it to 3 to find the value of k.