1. **Restate the problem:** Various mathematical expressions and graphs are mentioned including linear equations, cubic functions, factorization of polynomials, matrix representation, and trigonometric identities. 2. One important formula: $$\Psi_0(\vec{x})=c_1\psi_1(\vec{x})$$ This shows a wavefunction as an eigenstate. 3. For example, in the graph of a linear function: $$y=3x+5$$ - The slope is 3. - The y-intercept is 5. To find points, plug in x-values and calculate y. 4. For the cubic function $$y=x^3 - x^2 - 5x + 6,$$ substitute x-values to find y in order to plot the graph. This cubic has roots near $x=-2,1,3$. 5. Derivative of a cubic factorized form: Given $$\phi(t) = (t-p)(t-q)(t-r),$$ then $$\phi'(t) = (t-q)(t-r) + (t-p)(t-r) + (t-p)(t-q)$$ which is the sum of products of two linear terms each. 6. Matrix given: $$\begin{bmatrix} 2 & 5 & 8 \\ 4 & 7 & 2 \\ 1 & 6 & 9 \end{bmatrix}$$ represents numerical data or coefficients. 7. Equation of a line from parameters: $$Y = aX + b$$ where $$a = \frac{v l}{u (l-u)}, \quad b = \frac{h u}{l-u}$$ 8. Euler's identity raised to a power: $$(e^{i\pi}+1)^4 = 0$$ Since $e^{i\pi} = -1$, inside the parentheses is zero. 9. Limits and trigonometric identities: $$\lim_{x \to -\infty} e^x = 0$$ $$\cos^3 \theta = \frac{1}{4} \cos \theta + \frac{3}{4} \cos 3\theta$$ 10. Power series definition: $$f(z) = \sum_{n=0}^\infty a_n z^n$$ where $$k=\frac{2\pi}{\lambda}$$ **Summary:** The extraction covers multiple concepts: eigenstates in quantum mechanics, linear and cubic functions with graph points, polynomial derivatives, matrices, line parameters, Euler's formula, limits, trigonometric identities, and series expansions, providing a comprehensive mathematical review.