Question

If \[\left( x-1 \right)\] is a factor of \[{{x}^{2}}+2x-k\], then find \[k\].
A. \[-3\]
B. \[3\]
C. \[2\]
D. \[-2\]

Answer 1

Hint:To find the value of \[k\], we will use the factor theorem method, in this method we separate the factor that is given to us and equate it with zero to find the value of \[x\].

After getting the value of \[x\], we will put it in the equation given to us to find the value of \[k\] as shown below:
\[f(x)={{x}^{2}}+2x-k,\left( x-c=0 \right)\]
\[c\] is the value given to us.

Complete step by step solution:
First we will need the value of \[x\] that is given to us as \[\left( x-1 \right)\] and then we will equate it with zero to find the value of \[x\] as:

\[\Rightarrow x-1=0\]
\[\Rightarrow x=1\]

Now putting the value in the equation, we get the value of \[{{x}^{2}}+2x-k\] as:
\[\Rightarrow f(x)={{x}^{2}}+2x-k\]
\[\Rightarrow f(1)={{1}^{2}}+2\times 1-k\]
\[\Rightarrow {{1}^{2}}+2\times 1-k=0\]
\[\Rightarrow k=3\]

Hence, the value of \[k\] is given as \[3\].

Note: In factor theorem, if \[f\left( x \right)\] is the polynomial of degree \[n\ge 1\] and the other factor is given in terms of \[\left( x-c \right)\] with \[f\left( a \right)=0\]. The method is used for finding roots of polynomials if one of the roots is equal to zero.