Question

Degree 3. Zeros at

x = −5,
x = −2,
and
x = 1.
y-intercept at
(x, y) = (0, 5).

Answer 1

`\begin{cases} x = -5 &\implies x +5=0\\ x = -2 &\implies x +2=0\\ x = 1 &\implies x -1=0\\ \end{cases} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{original~polynomial}{a ( x +5 )( x +2 )( x -1 ) = \stackrel{0}{y}}\hspace{5em}\textit{we also know that} \begin{cases} x=0\\ y=5 \end{cases}`

`a ( 0 +5 )( 0 +2 )( 0 -1 ) = 5\implies -10a=5\implies a=\cfrac{5}{-10}\implies a=-\cfrac{1}{2} \\\\[-0.35em] ~\dotfill\\\\ -\cfrac{1}{2}( x +5 )( x +2 )( x -1 )=y\implies -\cfrac{1}{2}( x +5 )(x^2+x-2)=y \\\\\\ -\cfrac{1}{2}(x^3+6x^2+3x-10)=y\implies \boxed{-\cfrac{x^3}{2}-3x^2-\cfrac{3x}{2}+5=y}`