Question

How do you find k such that $f\left( x \right)={{x}^{4}}-k{{x}^{3}}+k{{x}^{2}}+1$ has the factor $x+2?$

Answer 1

Hint: The given polynomial in the question is $f\left( x \right)={{x}^{4}}-k{{x}^{3}}+k{{x}^{2}}+1.$ We know that this equation has a factor $x+2.$ This means that $x+2$ divides $f\left( x \right)$ completely. Therefore, $f\left( x \right)$ can be represented as
$f\left( x \right)=\left( x+2 \right).g\left( x \right)$
Where $g\left( x \right)$ is the term obtained by dividing $f\left( x \right)$ by $x+2.$

Complete step by step solution:
We know in general it is given if $x+a$ is a factor of the function $f\left( x \right),$ then $x=-a$ is a root of the equation $f\left( x \right)=0.$
We know that since $x+2$ is a factor of the equation, then $x=-2$ has to be a solution of the equation. Here, in $x=-2$ , -2 is called the root of the equation.
We also know that if $x=-a$ is a root of the equation then $f\left( -a \right)=0.$
Therefore, for the given question $x=-2$ is the root of the equation, $f\left( -2 \right)=0.$ Substituting for the value of x as -2 in the equation,
$\Rightarrow f\left( -2 \right)={{\left( -2 \right)}^{4}}-k{{\left( -2 \right)}^{3}}+k{{\left( -2 \right)}^{2}}+1$
Expanding the equation by calculating for different powers of -2 given,
$\Rightarrow f\left( -2 \right)=16-k\left( -8 \right)+4k+1$
We know that $f\left( -2 \right)=0,$therefore,
$\Rightarrow 16+8k+4k+1=0$
Adding the k terms and taking all the constant terms to right hand side,
$\Rightarrow 12k=-17$
Dividing both sides of the equation by 12,
$\Rightarrow k=\dfrac{-17}{12}$
Dividing -17 by 12 and simplifying to 4 decimal places,
$\Rightarrow k=-1.4166$
Hence the value k such that $f\left( x \right)={{x}^{4}}-k{{x}^{3}}+k{{x}^{2}}+1$ having the factor $x+2$ is -1.4166.

Note: To solve this question, students need to have a good understanding in the topics of factorization. By understanding these basic topics of factorization, we can solve these types of questions very easily. We need to note that we cannot solve this method using the long division method since all the coefficients of the function given are not known to us.