When analyzing a polynomial equation such as [tex]\( x^5 = -2x^2 \)[/tex]:
1. Rewrite the equation: Set all terms to one side of the equation to form a standard polynomial equation. This gives us:
[tex]\[
x^5 + 2x^2 = 0
\][/tex]
2. Factor the equation: To solve for [tex]\( x \)[/tex], factor out the common term. Here, both terms share a common factor of [tex]\( x^2 \)[/tex]:
[tex]\[
x^2(x^3 + 2) = 0
\][/tex]
3. Set each factor to zero: To find the roots of the equation, set each factor equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[
x^2 = 0 \quad \text{or} \quad x^3 + 2 = 0
\][/tex]
- Solving [tex]\( x^2 = 0 \)[/tex] gives:
[tex]\[
x = 0
\][/tex]
- Solving [tex]\( x^3 + 2 = 0 \)[/tex] gives:
[tex]\[
x^3 = -2 \implies x = \sqrt[3]{-2} \approx -1.2599
\][/tex]
4. Verify the number of distinct solutions: From the above, the roots of the polynomial equation are:
[tex]\[
x = 0 \quad \text{(a double root)} \quad \text{and} \quad x \approx -1.2599
\][/tex]
Thus, when Josh graphs the system of equations, he identifies two solutions. This is correct because the solutions to the equation [tex]\( x^5 = -2x^2 \)[/tex] include [tex]\( x = 0 \)[/tex] (which, despite being a double root, is counted as a single distinct intersection point) and [tex]\( x \approx -1.2599 \)[/tex], leading to a total of two points of intersection when viewed graphically. Therefore, his conclusion based on the graph is accurate.
