For the polynomial function below: (a) List each real zero and its multiplicity. (b) Determine
whether the graph crosses or touches the -axis at each -intercept. © Determine the
end behavior.

(a) Find any real zeros of .
The real zero(s) of is/are
The multiplicity of the smaller zero is
(Type a whole number.)
The multiplicity of the larger zero is
(Type a whole number.)
(b) The graph the -axis at the smaller -intercept.
The graph the -axis at the larger -intercept.
© Select the end behavior of the graph.
A. Falls left and rises right
B. Falls left and right
C. Rises left and right
D. Rises left and falls right

Ask by Curry Li. in the United States

Dec 12,2024

Question

For the polynomial function below: (a) List each real zero and its multiplicity. (b) Determine
whether the graph crosses or touches the -axis at each -intercept. © Determine the
end behavior.

(a) Find any real zeros of .
The real zero(s) of is/are
The multiplicity of the smaller zero is
(Type a whole number.)
The multiplicity of the larger zero is
(Type a whole number.)
(b) The graph the -axis at the smaller -intercept.
The graph the -axis at the larger -intercept.
© Select the end behavior of the graph.
A. Falls left and rises right
B. Falls left and right
C. Rises left and right
D. Rises left and falls right

Ask by Curry Li. in the United States

Dec 12,2024

UpStudy AI · Accepted Answer

Given polynomial function: $$ f(x) = (x-4)^2(x+2)^3 $$

(a) To find the real zeros of $$ f $$, we set $$ f(x) = 0 $$ and solve for $$ x $$:
$$ (x-4)^2(x+2)^3 = 0 $$

The real zeros of $$ f $$ are the values of $$ x $$ that make $$ f(x) = 0 $$. The real zeros are:
- $$ x = 4 $$ with multiplicity 2
- $$ x = -2 $$ with multiplicity 3

The multiplicity of the smaller zero (x = 4) is 2.
The multiplicity of the larger zero (x = -2) is 3.

(b) To determine whether the graph crosses or touches the $$ x $$-axis at each $$ x $$-intercept, we look at the multiplicity of the zeros:
- For the zero with multiplicity 2 (x = 4), the graph crosses the $$ x $$-axis.
- For the zero with multiplicity 3 (x = -2), the graph touches the $$ x $$-axis.

(c) To determine the end behavior of the graph, we look at the leading term of the polynomial function:
- The leading term is $$ x^5 $$ since the highest power of $$ x $$ is 5.
- Since the leading coefficient is positive, the end behavior is:
  - As $$ x $$ approaches negative infinity, $$ f(x) $$ falls to the left.
  - As $$ x $$ approaches positive infinity, $$ f(x) $$ rises to the right.

Therefore, the end behavior of the graph is:
- Falls left and rises right (Option A)
(a) Real zeros: $$ x = 4 $$ (multiplicity 2), $$ x = -2 $$ (multiplicity 3) (b) Graph crosses the $$ x $$-axis at $$ x = 4 $$. Graph touches the $$ x $$-axis at $$ x = -2 $$. (c) End behavior: Falls left and rises right

Solución de inteligencia artificial de Upstudy

Responder

Solución

Extra Insights

preguntas relacionadas

Latest Algebra Questions