To solve the problem of finding the real zeros of the polynomial function [tex]\( f(x) = x^3 - 12x^2 + 37x - 14 \)[/tex] and then using these zeros to factor the polynomial, we proceed as follows:
### Finding the Real Zeros
1. Identify the Polynomial: We start with the polynomial
[tex]\[
f(x) = x^3 - 12x^2 + 37x - 14
\][/tex]
2. Solve for the Zeros: To find the zeros, we solve the equation [tex]\( f(x) = 0 \)[/tex]:
[tex]\[
x^3 - 12x^2 + 37x - 14 = 0
\][/tex]
3. List the Real Zeros: Through careful solving or using appropriate algebraic techniques or tools, we determine that the real zeros are:
[tex]\[
x = 7, \quad x = \frac{5}{2} - \frac{\sqrt{17}}{2}, \quad \text{and} \quad x = \frac{5}{2} + \frac{\sqrt{17}}{2}
\][/tex]
### Factoring the Polynomial
1. Use the Zeros to Form Factors: Given the real zeros, we can express the polynomial as a product of its factors corresponding to each zero.
2. Form the Factors:
[tex]\[
f(x) = (x - 7)\left(x - \left(\frac{5}{2} - \frac{\sqrt{17}}{2}\right)\right)\left(x - \left(\frac{5}{2} + \frac{\sqrt{17}}{2}\right)\right)
\][/tex]
3. Simplify the Factored Form:
- Notice that the two complex conjugate factors can be combined into a quadratic:
[tex]\[
\left(x - \left(\frac{5}{2} - \frac{\sqrt{17}}{2}\right)\right)\left(x - \left(\frac{5}{2} + \frac{\sqrt{17}}{2}\right)\right)
\][/tex]
can be simplified using the difference of squares:
[tex]\[
= \left[\left(x - \frac{5}{2}\right)^2 - \left(\frac{\sqrt{17}}{2}\right)^2\right]
\][/tex]
[tex]\[
= \left[x^2 - 5x + \frac{25}{4}\right] - \frac{17}{4}
\][/tex]
[tex]\[
= x^2 - 5x + 2
\][/tex]
4. Write the Final Factored Form:
[tex]\[
f(x) = (x - 7)(x^2 - 5x + 2)
\][/tex]
### Summary of Results
- Real Zeros:
[tex]\[
x = 7, \quad x = \frac{5}{2} - \frac{\sqrt{17}}{2}, \quad x = \frac{5}{2} + \frac{\sqrt{17}}{2}
\][/tex]
- Factored Form:
[tex]\[
f(x) = (x - 7)(x^2 - 5x + 2)
\][/tex]
The real zeros of [tex]\( f(x) \)[/tex] are [tex]\( 7, \frac{5}{2} - \frac{\sqrt{17}}{2}, \frac{5}{2} + \frac{\sqrt{17}}{2} \)[/tex], and the polynomial [tex]\( f(x) \)[/tex] can be factored as [tex]\( (x - 7)(x^2 - 5x + 2) \)[/tex].
