Question

Find the value of k for which (x-1) is a factor of
(2x^3+9x²+x+k)​

Answer 1

X-1=0

x=1

Putting the value of X

2(1)^3+9(1)^2+1+k

2+9+1+k=0

12+k=0

k=-12

Answer 2

Answer:

k = -12

Note:

★ Remainder theorem : If a polynomial p(x) is divided by (x - a) , then the remainder is given as , R = p(a).

★ Factor theorem : i) If (x - a) is a factor of the polynomial p(x) , then the remainder obtained on dividing p(x) by (x - a) is zero , ie ; R = p(a) = 0.

ii) If the remainder obtained on dividing the polynomial p(x) by (x - a) is zero , ie ; if p(a) = 0 , then (x - a) is a factor of p(x).

Solution:

Let the given polynomial be p(x) .

Thus,

p(x) = 2x³ + 9x² + x + k

Also,

It is given that , (x - 1) is a factor of the given polynomial p(x).

Thus,

According to the factor theorem , the remainder obtained on dividing p(x) by (x - 1) must be zero .

Thus,

=> R = 0

=> p(1) = 0

=> 2•(1)² + 9•(1)² + (1) + k = 0

=> 2 + 9 + 1 + k = 0

=> 12 + k = 0

=> k = -12

Hence,

The required value of k is (-12) .