For the polynomial function $$ 1(x)=x^{4}+4 x^{3}+4 x^{2} $$, enswer the parts a through e. a. Use the Leading Coofficient Test to delermine the graph's end behavior. A. The graph of $$ f(x) $$ falls to the left and falls to the right. B. The graph of $$ f(x) $$ rises to the lelt and falls to the right C. The graph of $$ f(x) $$ rises to the lelt and rises to the right. D. The graph of $$ f(x) $$ falls to the lefl and rises to the right. b. Find the $$ x $$-intercapt(5). Stato whether the graph crosses tho $$ x $$-axis, or touches the $$ x $$-axis and turns around, at each inter The $$ x $$-intercept(s) is/are $$ \square $$ ]. (Type an integer or a decimal. Uso a comma lo senarale answers as needed. Type each answer only once.)

Question

UpStudy AI · Accepted Answer

**a. Leading Coefficient Test**

The polynomial is  
$$
f(x)=x^4+4 x^3+4 x^2.
$$

First, note that this polynomial is degree 4 (an even degree) and its leading coefficient is 1 (positive). The Leading Coefficient Test tells us that when the degree of a polynomial is even and the leading coefficient is positive, the graph rises on both ends.

Thus, the correct answer is:  
**C. The graph of $$ f(x) $$ rises to the left and rises to the right.**

---

**b. Finding the $$ x $$-intercepts**

To find the $$ x $$-intercepts of $$ f(x) $$, set $$ f(x)=0 $$:

$$
x^4+4 x^3+4 x^2=0.
$$

Factor out the common term $$ x^2 $$:

$$
x^2 (x^2+4 x+4)=0.
$$

Recognize that the quadratic is a perfect square:

$$
x^2+4x+4=(x+2)^2.
$$

Thus, we can write

$$
x^2 (x+2)^2=0.
$$

Setting each factor equal to zero gives:
- $$ x^2=0 $$ which implies $$ x=0 $$,
- $$ (x+2)^2=0 $$ which implies $$ x+2=0 $$ or $$ x=-2 $$.

Since both factors are squared (even multiplicity), at each $$ x $$-intercept the graph touches the $$ x $$-axis and turns around.

Thus, the $$ x $$-intercepts are:  
$$
x = -2 \quad \text{and} \quad x = 0.
$$
**a. Leading Coefficient Test** - The graph of $$ f(x) $$ rises to the left and rises to the right. **b. $$ x $$-intercepts** - The $$ x $$-intercepts are $$ x = -2 $$ and $$ x = 0 $$.

Upstudy AI Solution

Answer

Solution

The Deep Dive

Related Questions

Latest Algebra Questions