Let $f ( x ) = a x ^ { 3 } + b x ^ { 2 } + c x + d , a \neq 0$ and $g ( x ) = k x ^ { 4 } + l x ^ { 3 } + m x ^ { 2 } + n x + p, k \neq 0 .$ Hint: Use the end behavior of polynomials and the Intermediate Value Theorem.) Show that f has exactly one real zero or three real zeros counting multiplicities.

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We have a cubic polynomial

f(x) = ax^3 + bx^2 + cx + d, \ \ a \ne 0

Now as it is cubic the end behavior of the polynomial will of opposite sign that is if as $x \to \infty \ \Rightarrow f(x) \to \infty$ then $x \to - \infty \ \Rightarrow f(x) \to - \infty$ or vice versa. with this we will get real numbers $l$ and $m$ such that $f(l)$ and $f(m)$ will have different signs so by Intermediate Value Theorem we have at least one $x$ -intercept, a real root to $f$ . Now there are cases $\textbf{Case I }$ In this case the curve cross $x$ -axis exactly one time, so there is exactly one real root to $f$ $\textbf{Case II }$ In this case the curve cross $x$ -axis more than one time, now due to end behavior it has to cross the $x$ -axis 3 times(it can not cross more than 3 as degree is 3 so maximum turns are 2) so in this case there are exactly 3 real roots to $f$