Answer
a. The polynomial function's graph rises to the left end and falls to the right end.
b. Graph $V$
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, $R(x)=-x^5+5x^3-4x$, the function is of odd degree and negative leading coefficient. Therefore, the polynomial function's graph rises to the left end and falls to the right end.
-Solving for the polynomial, $R(x)=-x^5+5x^3-4x$,
Taking out $-x$.
$-x(x^4-5x^2+4)=0$,
thus, either $-x=0, x=0$ or $x^4-5x^2+4=0$,
-solving for the trinomial,
lets let $x^2=k$,
$k^2-5k+4$,
$k^2-k-4k+4$.
$k(k-1)-4(k-1)$,
$(k-4)(k-1)$,
substituting $x^2=k$,
$(x^2-4)(x^2-1)$,
factoring out completely...
$(x-2)(x+2)(x-1)(x+1)$,
Thus, $R(x)=-x(x-2)(x+2)(x-1)(x+1)$,
from the $R(x)$ factors we can findout that $R(x)$ has a zero of $x=\{0, -2, -1, 1, 2\}$ all with a multiplicity of $1$.
Therefore, $R(x)$ crosses $x-axis$ at all zeros.
b. The graph that matches the description is Graph $V$
