Answer :

To find all zeros of the function [tex]\( f(x) = 3x^3 + 19x^2 + 15x - 25 \)[/tex], follow these steps:

1. Identify the polynomial function:
[tex]\[
f(x) = 3x^3 + 19x^2 + 15x - 25
\][/tex]

2. Determine possible rational roots:
According to the Rational Root Theorem, the possible rational roots are the factors of the constant term (-25) divided by the factors of the leading coefficient (3). The factors of -25 are [tex]\(\pm 1, \pm 5, \pm 25\)[/tex], and the factors of 3 are [tex]\(\pm 1, \pm 3\)[/tex]. So the possible rational roots are:
[tex]\[
\pm 1, \pm 5, \pm 25, \pm \frac{1}{3}, \pm \frac{5}{3}, \pm \frac{25}{3}
\][/tex]

3. Test possible rational roots:
Let's test these in the polynomial to see if they yield zero.

[tex]\[
f(-5) = 3(-5)^3 + 19(-5)^2 + 15(-5) - 25 = 3(-125) + 19(25) + 15(-5) - 25 = -375 + 475 - 75 - 25 = 0
\][/tex]
So, [tex]\( x = -5 \)[/tex] is a root of the polynomial [tex]\( f(x) \)[/tex].

4. Factor out the found root:
We know [tex]\( x = -5 \)[/tex] is a root, so we can divide [tex]\( f(x) \)[/tex] by [tex]\( (x + 5) \)[/tex] to find the other factors. This can be done using polynomial long division or synthetic division.

5. Solve the quadratic equation:
After dividing [tex]\( f(x) \)[/tex] by [tex]\( (x + 5) \)[/tex], we get:
[tex]\[
f(x) = (x + 5)(3x^2 + 4x - 5)
\][/tex]
Now solve the quadratic [tex]\( 3x^2 + 4x - 5 = 0 \)[/tex] using the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Here, [tex]\( a = 3 \)[/tex], [tex]\( b = 4 \)[/tex], and [tex]\( c = -5 \)[/tex].

[tex]\[
x = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 3 \cdot (-5)}}{2 \cdot 3}
= \frac{-4 \pm \sqrt{16 + 60}}{6}
= \frac{-4 \pm \sqrt{76}}{6}
= \frac{-4 \pm 2\sqrt{19}}{6}
= \frac{-4}{6} \pm \frac{2\sqrt{19}}{6}
= \frac{-2}{3} \pm \frac{\sqrt{19}}{3}
\][/tex]
So, the roots are:
[tex]\[
x = \frac{-2 + \sqrt{19}}{3}
\][/tex]
and
[tex]\[
x = \frac{-2 - \sqrt{19}}{3}
\][/tex]

6. List all the zeros:
The zeros of the function [tex]\( f(x) = 3x^3 + 19x^2 + 15x - 25 \)[/tex] are:
[tex]\[
x = -5, \quad x = \frac{-2 + \sqrt{19}}{3}, \quad x = \frac{-2 - \sqrt{19}}{3}
\][/tex]

These are the zeros of the function.

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Rewritten by : Barada