Question

Factor the polynomial function f(15x^3-7x^2-27x-5.

Note: Write (x- a/b) as (bx-a)


The zeros are 5/3, -1/5, -1
f(x)=

Answer 1

To factor the polynomial \( f(x) = 15x^3 - 7x^2 - 27x - 5 \) given the zeros \( \frac{5}{3} \), \( -\frac{1}{5} \), and \( -1 \), we can use the fact that if \( r \) is a zero of the polynomial, then \( (x - r) \) is a factor.

1. **Finding the factors:**

- The zero \( \frac{5}{3} \) gives the factor \( (3x - 5) \).
- The zero \( -\frac{1}{5} \) gives the factor \( (5x + 1) \).
- The zero \( -1 \) gives the factor \( (x + 1) \).

2. **Form the polynomial from the factors:**

Therefore, we can represent the polynomial as:
\[
f(x) = k(3x - 5)(5x + 1)(x + 1)
\]
where \( k \) is a constant.

3. **Expanding the product:**

First, let's expand \( (3x - 5)(5x + 1) \):
\[
(3x - 5)(5x + 1) = 15x^2 + 3x - 25x - 5 = 15x^2 - 22x - 5.
\]

Now, multiply this by \( (x + 1) \):
\[
(15x^2 - 22x - 5)(x + 1) = 15x^3 + 15x^2 - 22x^2 - 22x - 5x - 5 = 15x^3 - 7x^2 - 27x - 5.
\]

4. **Identifying the constant \( k \):**

Since the coefficients match perfectly with the original polynomial \( f(x) \), we can say \( k = 1 \).

5. **Final expression of the factored form:**

Therefore, the factored form of the polynomial \( f(x) \) is:
\[
f(x) = (3x - 5)(5x + 1)(x + 1).
\]

Thus, the final answer is:
\[
\boxed{(3x - 5)(5x + 1)(x + 1)}.
\]