Answer
a. The polynomial function's graph rises to the left end and rises to the right end.
b. Graph $VI$
Work Step by Step
End-Behaviour of a Polynomial
- Polynomial function of odd degree and positive leading coefficient falls to the left end and rises to the right end.
- Polynomial function of odd degree and negative leading coefficient rises to the left end and falls to the right end.
- Polynomial function of even degree and positive leading coefficient rises to the left end and right end.
- Polynomial function of even degree and negative leading coefficient falls to the left end and right end.
a.
In this case, $T(x)=x^4+2x^3$, the function is of even degree and positive leading coefficient. therefore, the polynomial function's graph rises to the left end and rises to the right end.
- Solving for the polynomial, $T(x)=x^4+2x^3$,
Taking out $x^3$.
$x^3(x+2)=0$,
thus, either $x^3=0, x=0$ or $x+2=0, x=-2$,
Therefore, $T(x)=x^3(x+2)$
from the $T(x)$ factors we can findout that $T(x)$ has a zero of $x=\{0\}$ with a multiplicity of $3$ and $x=\{-2\}$ with a multiplicity of $1$.
Therefore, $T(x)$ crosses negative $x-axis$ at $x=-2$ and crosses $x-axis$ at $x=0$.
b. The graph that matches the description is Graph $VI$
