Activity 5.2.2.
By experimenting with coefficients in Desmos, find a formula for a polynomial function that has the stated properties, or explain why no such polynomial exists. If you enter
p(x)=a+bx+cx^2+dx^3+fx^4+gx^5 in Desmos, you’ll get prompted to add sliders that make it easy to explore a degree \(5\) polynomial. (We skip using e as one of the constants since Desmos reserves e as the Euler constant.)
(a)
A polynomial \(p\) of degree \(5\) with exactly \(3\) real zeros, \(4\) turning points, and such that \(\lim_{x \to -\infty} p(x) = +\infty\) and \(\lim_{x \to \infty} p(x) = -\infty\text{.}\)
(b)
A polynomial \(p\) of degree \(4\) with exactly \(4\) real zeros, \(3\) turning points, and such that \(\lim_{x \to -\infty} p(x) = +\infty\) and \(\lim_{x \to \infty} p(x) = -\infty\text{.}\)
(c)
A polynomial \(p\) of degree \(6\) with exactly \(2\) real zeros, \(3\) turning points, and such that \(\lim_{x \to -\infty} p(x) = -\infty\) and \(\lim_{x \to \infty} p(x) = -\infty\text{.}\)
(d)
A polynomial \(p\) of degree \(5\) with exactly \(5\) real zeros, \(3\) turning points, and such that \(\lim_{x \to -\infty} p(x) = +\infty\) and \(\lim_{x \to \infty} p(x) = -\infty\text{.}\)
