To find the [tex]\( x \)[/tex]-intercepts of the polynomial [tex]\( f(x) = (2x - 3)(x - 4)(x + 3) \)[/tex], we need to determine the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 0 \)[/tex]. These values are also known as the roots or solutions of the polynomial equation.
1. Set the polynomial [tex]\( f(x) \)[/tex] to zero:
[tex]\[
(2x - 3)(x - 4)(x + 3) = 0
\][/tex]
2. Apply the Zero Product Property:
The Zero Product Property states that if a product of factors is zero, at least one of the factors must be zero. Therefore, we set each factor of the polynomial to zero and solve for [tex]\( x \)[/tex].
- For the factor [tex]\( 2x - 3 \)[/tex]:
[tex]\[
2x - 3 = 0
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
2x = 3
\][/tex]
[tex]\[
x = \frac{3}{2}
\][/tex]
- For the factor [tex]\( x - 4 \)[/tex]:
[tex]\[
x - 4 = 0
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
x = 4
\][/tex]
- For the factor [tex]\( x + 3 \)[/tex]:
[tex]\[
x + 3 = 0
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
x = -3
\][/tex]
3. List the [tex]\( x \)[/tex]-intercepts:
The [tex]\( x \)[/tex]-intercepts are the values of [tex]\( x \)[/tex] that make the polynomial equal to zero. These values are:
[tex]\[
x = -3, \quad x = \frac{3}{2}, \quad x = 4
\][/tex]
Thus, the graph of the polynomial [tex]\( f(x) = (2x - 3)(x - 4)(x + 3) \)[/tex] has [tex]\( x \)[/tex]-intercepts at [tex]\( x = -3 \)[/tex], [tex]\( x = \frac{3}{2} \)[/tex], and [tex]\( x = 4 \)[/tex].
