To find a polynomial [tex]\( f(x) \)[/tex] with integer coefficients that has the given zeros [tex]\( -\frac{5}{3} \)[/tex], [tex]\( \frac{3}{5} \)[/tex], and [tex]\( 6 \)[/tex], follow these steps:
1. Express each zero as a root of a factor:
- The root [tex]\( -\frac{5}{3} \)[/tex] corresponds to the factor [tex]\( \left(3x + 5\right) \)[/tex].
- The root [tex]\( \frac{3}{5} \)[/tex] corresponds to the factor [tex]\( \left(5x - 3\right) \)[/tex].
- The root [tex]\( 6 \)[/tex] corresponds to the factor [tex]\( \left(x - 6\right) \)[/tex].
2. Write the polynomial as the product of the factors:
Because we want the polynomial to have integer coefficients, we can multiply these factors together to form our polynomial:
[tex]\[
f(x) = (3x + 5)(5x - 3)(x - 6)
\][/tex]
3. Expand the polynomial:
Let's first expand the product of the first two factors:
[tex]\[
(3x + 5)(5x - 3)
\][/tex]
Distribute each term in [tex]\( 3x + 5 \)[/tex]:
[tex]\[
(3x + 5)(5x - 3) = 3x(5x) + 3x(-3) + 5(5x) + 5(-3)
\][/tex]
[tex]\[
= 15x^2 - 9x + 25x - 15
\][/tex]
Combine like terms:
[tex]\[
= 15x^2 + 16x - 15
\][/tex]
Now, multiply this resulting polynomial by the third factor:
[tex]\[
(15x^2 + 16x - 15)(x - 6)
\][/tex]
Distribute each term in [tex]\( 15x^2 + 16x - 15 \)[/tex]:
[tex]\[
= 15x^2(x) + 15x^2(-6) + 16x(x) + 16x(-6) - 15(x) - 15(-6)
\][/tex]
[tex]\[
= 15x^3 - 90x^2 + 16x^2 - 96x - 15x + 90
\][/tex]
Combine like terms:
[tex]\[
15x^3 + (16x^2 - 90x^2) - 96x - 15x + 90
\][/tex]
[tex]\[
= 15x^3 - 74x^2 - 111x + 90
\][/tex]
4. Write the final polynomial:
[tex]\[
f(x) = 15x^3 - 74x^2 - 111x + 90
\][/tex]
So, the polynomial [tex]\( f(x) \)[/tex] that meets the given conditions is:
[tex]\[
f(x) = 15x^3 - 74x^2 - 111x + 90
\][/tex]
