Write the polynomial [tex][tex]$f(x)$[/tex][/tex] that meets the given conditions. Answers may vary.

Degree 3 polynomial with integer coefficients and zeros of [tex][tex]$-\frac{5}{3}$[/tex][/tex], [tex][tex]$\frac{3}{5}$[/tex][/tex], and 6.



Answer :

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]