Answer

Verified

413.7k+ views

**Hint:**In this problem we have to calculate the roots of the given equation. We can observe that the given equation is the cubic equation. So, we will solve this problem by using the factors of the cubic equation. Now we will consider the first two terms individually and take ${{x}^{2}}$ as common. After that we will consider the last two terms and take $-16$ as common. Now we will observe the obtained equation and take appropriate terms as common to get the factors of the given equation. After getting the factors of the cubic equation, we will equate each factor to zero and calculate the values of $x$ which are our required roots.

**Complete step by step solution:**

Given equation, $f\left( x \right)={{x}^{3}}-4{{x}^{2}}-16x+64$.

Considering the first two terms. We have the first term ${{x}^{3}}$ and the second term $-4{{x}^{2}}$. By observing the above two terms we can take ${{x}^{2}}$ as common. So, taking ${{x}^{2}}$ as common from the first two terms of the given equation, then we will get

$\Rightarrow {{x}^{3}}-4{{x}^{2}}-16x+64={{x}^{2}}\left( x-4 \right)-16x+64$

Considering the last two terms of the given equation. We have last term $64$ and the third term $-16x$. By observing the above two terms we can take $-16$ as common. So, taking $-16$ as common from the last two terms of the given equation, then we will get

$\Rightarrow {{x}^{3}}-4{{x}^{2}}-16x+64={{x}^{2}}\left( x-4 \right)-16\left( x-4 \right)$

In the above equation we can observe that we can take $x-4$ as common. So, taking $x-4$ as common from the above equation, then we will get

$\Rightarrow {{x}^{3}}-4{{x}^{2}}-16x+64=\left( x-4 \right)\left( {{x}^{2}}-16 \right)$

Hence the factors of the given equation $f\left( x \right)={{x}^{3}}-4{{x}^{2}}-16x+64$ are ${{x}^{2}}-16$, $x-4$.

Considering the first factor which is ${{x}^{2}}-16$. Equating this factor to zero, then we will get

$\Rightarrow {{x}^{2}}-16=0$

We can write $16={{4}^{2}}$, then we will have

$\Rightarrow {{x}^{2}}-{{4}^{2}}=0$

Applying the formula ${{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)$ in the above equation, then we will have

$\Rightarrow \left( x+4 \right)\left( x-4 \right)=0$

Equating each term individually to zero, then we will get

$\begin{align}

& \Rightarrow x+4=0\text{ or }x-4=0 \\

& \Rightarrow x=-4\text{ or }x=4 \\

\end{align}$

Considering the second factor $x-4$. Equating this factor to zero, then we will have

$\begin{align}

& \Rightarrow x-4=0 \\

& \Rightarrow x=4 \\

\end{align}$

**Hence the roots of the given equation $f\left( x \right)={{x}^{3}}-4{{x}^{2}}-16x+64$ are $\pm 4$.**

**Note:**For this problem we have the factor ${{x}^{2}}-16$ which is in form of ${{a}^{2}}-{{b}^{2}}$, so we have used the algebraic formula and equated it to zero. In many cases we may get a quadratic equation which is in the form of $a{{x}^{2}}+bx+c$. Now we will use the formula $x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ and calculate the roots of the equation.

Recently Updated Pages

what is the correct chronological order of the following class 10 social science CBSE

Which of the following was not the actual cause for class 10 social science CBSE

Which of the following statements is not correct A class 10 social science CBSE

Which of the following leaders was not present in the class 10 social science CBSE

Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE

Which one of the following places is not covered by class 10 social science CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

Who was the Governor general of India at the time of class 11 social science CBSE

How do you graph the function fx 4x class 9 maths CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

One Metric ton is equal to kg A 10000 B 1000 C 100 class 11 physics CBSE

In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Why is there a time difference of about 5 hours between class 10 social science CBSE

Difference Between Plant Cell and Animal Cell