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Given:
The function is
[tex]f(x)=2x^3+kx-9[/tex]
The remainder when f(x) is divided by x−2 is 23.
To find:
The value of k.
Solution:
According to the remainder theorem, if a polynomial P(x) is divided by (x-c), then the remainder is P(c).
It is given that, the remainder when f(x) is divided by x−2 is 23. By using remainder theorem, we get
[tex]f(2)=23[/tex] ...(i)
Put x=2, to find the value of f(2).
[tex]f(2)=2(2)^3+k(2)-9[/tex]
[tex]f(2)=2(8)+2k-9[/tex]
[tex]f(2)=16+2k-9[/tex]
[tex]f(2)=2k+7[/tex] ...(ii)
Using (i) and (ii), we get
[tex]2k+7=23[/tex]
[tex]2k=23-7[/tex]
[tex]k=\dfrac{16}{2}[/tex]
[tex]k=8[/tex]
Therefore, the value of k is 8.