Sure, let's work through the polynomial equation step-by-step.
Firstly, the polynomial given to us is:
[tex]\[ f(x) = x^5 - 2x^2 \][/tex]
To find the roots of this polynomial, we need to set the equation equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ x^5 - 2x^2 = 0 \][/tex]
Next, we factor the polynomial. Notice that both terms on the left-hand side share a common factor of [tex]\( x^2 \)[/tex]. So we can factor out [tex]\( x^2 \)[/tex] from each term:
[tex]\[ x^2 (x^3 - 2) = 0 \][/tex]
Now, we set each factor equal to zero and solve for [tex]\( x \)[/tex].
First, for the factor [tex]\( x^2 \)[/tex]:
[tex]\[ x^2 = 0 \][/tex]
Taking the square root of both sides, we get:
[tex]\[ x = 0 \][/tex]
This root has a multiplicity of 2, meaning it is counted twice as it is derived from [tex]\( x^2 \)[/tex].
Secondly, for the factor [tex]\( x^3 - 2 \)[/tex]:
[tex]\[ x^3 - 2 = 0 \][/tex]
Adding 2 to both sides, we have:
[tex]\[ x^3 = 2 \][/tex]
Taking the cube root of both sides, we get:
[tex]\[ x = 2^{1/3} \][/tex]
The cube root of 2 is approximately 1.2599210498948732.
Thus, there are indeed two distinct real solutions to the polynomial equation [tex]\( x^5 - 2x^2 = 0 \)[/tex]. They are:
1. [tex]\( x = 0 \)[/tex]
2. [tex]\( x = 2^{1/3} \)[/tex] (approximately 1.2599210498948732)
When the polynomial is graphed, these are the points where the graph intersects the x-axis. Hence, Josh is correct in stating that the graph shows two intersection points.
