Question

What is the solution set for \[3{{x}^{5}}-48x=0\]?

Answer 1

Hint: We are given a question in which we have an expression and we have to find the solution of the given expression. We will solve the given expression to write the expression in terms of ‘x’, thereby finding the value of ‘x’. So, we will start by taking the common terms out from the given expression and then we will reduce the residual expression further using the algebraic identity, \[{{x}^{2}}-{{y}^{2}}=\left( x+y \right)\left( x-y \right)\]. Then, we will solve the expression further and equate all the simplified terms equal to 0 and we will have the solution of the given expression.

Complete step-by-step answer:
According to the given question, we are given an expression which we have to solve for ‘x’.
The expression we have is,
\[3{{x}^{5}}-48x=0\]
We will first take the common terms out and we get,
\[\Rightarrow 3x\left( {{x}^{4}}-16 \right)=0\]
So, we get the expression as,
\[\Rightarrow 3x=0\] or \[\left( {{x}^{4}}-16 \right)=0\]
For \[3x=0\], we have \[x=0\]
And for \[{{x}^{4}}-16=0\], we get,
\[\Rightarrow {{\left( {{x}^{2}} \right)}^{2}}-{{4}^{2}}=0\]
We can now use the algebraic identity, \[{{x}^{2}}-{{y}^{2}}=\left( x+y \right)\left( x-y \right)\] and we get,
\[\Rightarrow \left( {{x}^{2}}+4 \right)\left( {{x}^{2}}-4 \right)=0\]
Splitting up the terms further using the same identity, \[{{x}^{2}}-{{y}^{2}}=\left( x+y \right)\left( x-y \right)\], we get,
\[\Rightarrow \left( {{x}^{2}}+4 \right)\left( {{x}^{2}}-{{2}^{2}} \right)=0\]
\[\Rightarrow \left( {{x}^{2}}+4 \right)\left( x+2 \right)\left( x-2 \right)=0\]
Now, we will equate each of the terms to 0, we have,
\[{{x}^{2}}+4=0\] which we can write it as, \[{{x}^{2}}=-4\], then,
\[\Rightarrow {{x}^{2}}=4{{i}^{2}}\]
As we know that, \[{{i}^{2}}=-1\], we get,
\[\Rightarrow x=\pm 2i\]
Next, we have,
\[x+2=0\]
We get, \[\Rightarrow x=-2\]
Next, we have,
\[x-2=0\]
We get the value of ‘x’ as,
\[\Rightarrow x=2\]
Therefore, the solutions of the given expression are \[0,\pm 2i,\pm 2\].

Note: The point to be noted here is that the degree of a polynomial tells the number of solutions that particular can have. Degree of the polynomial refers to the highest power in the given polynomial. Here, in the given expression we have the highest power of the polynomial as 5, so the polynomial has 5 solutions or 5 zeroes of the polynomial exist, three are real and two imaginary.