Question

Find a polynomial function f(x) of least possible degree having the graph shown

Answer 1

first off let's notice a few things on that function.

the graph "passes" -5, that's a root, the graph "touches" 3 and goes back up, that's another root, but that's a root with an "even multiplicity", the heck does that mean? well, it means there are at least two roots, could be 4 or 6 or 18, so long is even, but we're shooting for the least degree, so we'll settle for 2.

now hmm, let's reword that

what's the equation of a function with roots at -5 and 3 twice, that it passes through (0 , 9)?

`\begin{cases} x = -5 &\implies x +5=0\\ x = 3 &\implies x -3=0\\ x = 3 &\implies x -3=0\\ \end{cases} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{original~polynomial}{a ( x +5 )( x -3 )( x -3 ) = \stackrel{0}{y}}\hspace{5em}\textit{we also know that } \begin{cases} x=0\\ y=9 \end{cases} \\\\\\ a ( 0 +5 )( 0 -3 )( 0 -3 ) = 9\implies 45a=9\implies a=\cfrac{9}{45}\implies a=\cfrac{1}{5} \\\\[-0.35em] ~\dotfill`

`\cfrac{1}{5}(x+5)(x-3)^2=y\implies \cfrac{1}{5}(x+5)(x^2-6x+9)=y \\\\\\ \cfrac{1}{5}(x^3-x^2-21x+45)=y\implies {\Large \begin{array}{llll} \cfrac{x^3}{5}-\cfrac{x^2}{5}-\cfrac{21x}{5}+9=y \end{array}}`

Check the picture below.