To find the zeros of the polynomial [tex]\( p(x) = (x^2 + 4x + 3)(x^2 - 4) \)[/tex], we need to determine the values of [tex]\( x \)[/tex] for which [tex]\( p(x) = 0 \)[/tex]. This can be done by setting each factor of the polynomial to zero and solving for [tex]\( x \)[/tex].
The polynomial is given by:
[tex]\[
p(x) = (x^2 + 4x + 3)(x^2 - 4)
\][/tex]
### Step-by-Step Solution:
#### Step 1: Solve [tex]\( x^2 + 4x + 3 = 0 \)[/tex]
This is a quadratic equation. To solve it, we can use the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
For the quadratic [tex]\( x^2 + 4x + 3 \)[/tex], the coefficients are [tex]\( a = 1 \)[/tex], [tex]\( b = 4 \)[/tex], and [tex]\( c = 3 \)[/tex]. Plugging these into the quadratic formula:
[tex]\[
x = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 1 \cdot 3}}{2 \cdot 1} = \frac{-4 \pm \sqrt{16 - 12}}{2} = \frac{-4 \pm \sqrt{4}}{2} = \frac{-4 \pm 2}{2}
\][/tex]
This yields two solutions:
[tex]\[
x = \frac{-4 + 2}{2} = \frac{-2}{2} = -1 \quad \text{and} \quad x = \frac{-4 - 2}{2} = \frac{-6}{2} = -3
\][/tex]
So, [tex]\( x = -1 \)[/tex] and [tex]\( x = -3 \)[/tex] are zeros of the factor [tex]\( x^2 + 4x + 3 \)[/tex].
#### Step 2: Solve [tex]\( x^2 - 4 = 0 \)[/tex]
This is another quadratic equation that can be factored directly as:
[tex]\[
x^2 - 4 = (x - 2)(x + 2)
\][/tex]
Setting each factor to zero, we get:
[tex]\[
x - 2 = 0 \quad \Rightarrow \quad x = 2
\][/tex]
[tex]\[
x + 2 = 0 \quad \Rightarrow \quad x = -2
\][/tex]
So, [tex]\( x = 2 \)[/tex] and [tex]\( x = -2 \)[/tex] are zeros of the factor [tex]\( x^2 - 4 \)[/tex].
### Conclusion:
The zeros of the polynomial [tex]\( p(x) = (x^2 + 4x + 3)(x^2 - 4) \)[/tex] are [tex]\( x = -3 \)[/tex], [tex]\( x = -2 \)[/tex], [tex]\( x = -1 \)[/tex], and [tex]\( x = 2 \)[/tex].
### Plotting the Zeros:
Let's plot the zeros on the [tex]\( x \)[/tex]-axis. The zeros are the points where the polynomial intersects the [tex]\( x \)[/tex]-axis.
```
Zero | Coordinate
-------------------
-3 | (-3, 0)
-2 | (-2, 0)
-1 | (-1, 0)
2 | (2, 0)
```
[tex]\[
\begin{tikzpicture}
\begin{axis}[
axis lines = middle,
xlabel = $x$,
ylabel = $y$,
ymin = -1.5, ymax = 1.5,
xmin = -4, xmax = 3,
ytick={\empty},
xtick={-3,-2,-1,2}
]
% Plot the points
\addplot[mark=*] coordinates {(-3,0)(-2,0)(-1,0)(2,0)};
\end{axis}
\end{tikzpicture}
\][/tex]
In a graph, the zeros of the polynomial [tex]\( p(x) \)[/tex] are highlighted as points [tex]\((-3,0)\)[/tex], [tex]\((-2,0)\)[/tex], [tex]\((-1,0)\)[/tex], and [tex]\((2,0)\)[/tex].
This interactive plot shows the locations of the zeros on the [tex]\( x \)[/tex]-axis.
