Question

Find the x-intercepts of the polynomial function k(x)=4x(1−2x)(3x+4)

Answer 1

To find the x-intercepts of the polynomial function \( k(x) = 4x(1 - 2x)(3x + 4) \), we need to set the function equal to zero and solve for \( x \):

\[
k(x) = 0
\]

This means we need to solve:

\[
4x(1 - 2x)(3x + 4) = 0
\]

The product of three factors equals zero if at least one of the factors equals zero. Therefore, we set each factor equal to zero:

1. \( 4x = 0 \)
2. \( 1 - 2x = 0 \)
3. \( 3x + 4 = 0 \)

Now, let's solve each equation:

1. For \( 4x = 0 \):
\[
x = 0
\]

2. For \( 1 - 2x = 0 \):
\[
2x = 1 \quad \Rightarrow \quad x = \frac{1}{2}
\]

3. For \( 3x + 4 = 0 \):
\[
3x = -4 \quad \Rightarrow \quad x = -\frac{4}{3}
\]

Thus, the x-intercepts of the polynomial function \( k(x) \) are:

\[
x = 0, \quad x = \frac{1}{2}, \quad x = -\frac{4}{3}
\]

In conclusion, the x-intercepts of the function \( k(x) = 4x(1 - 2x)(3x + 4) \) are:

\[
\boxed{0, \frac{1}{2}, -\frac{4}{3}}
\]