Find a polynomial function of degree 3 with real coefficients that has the given zeros.

minus1​, 

2​,minus4




The polynomial function can be constructed using the given zeros. If a polynomial has zeros \( x = -1 \), \( x = 2 \), and \( x = -4 \), we can express the polynomial in terms of its factors. 

For each zero, there is a corresponding factor of the form \( (x - r) \), where \( r \) is the zero.

- For the zero \( x = -1 \), the factor is \( (x + 1) \).
- For the zero \( x = 2 \), the factor is \( (x - 2) \).
- For the zero \( x = -4 \), the factor is \( (x + 4) \).

Thus, the polynomial is:
\[
f(x) = (x + 1)(x - 2)(x + 4)
\]

Now, let's expand the expression:
\[
f(x) = (x + 1)(x^2 + 4x - 2x - 8) = (x + 1)(x^2 + 2x - 8)
\]
\[
f(x) = x(x^2 + 2x - 8) + 1(x^2 + 2x - 8)
\]
\[
f(x) = x^3 + 2x^2 - 8x + x^2 + 2x - 8
\]
\[
f(x) = x^3 + 3x^2 - 6x - 8
\]

Therefore, the polynomial function is:
\[
f(x) = x^3 + 3x^2 - 6x - 8
\]

Would you like any further details or explanations?

Here are 5 related questions:
1. How can you find the degree of a polynomial from its factors?
2. What does it mean for a polynomial to have real coefficients?
3. How do you verify that the given zeros are correct for this polynomial?
4. How do you factor polynomials of higher degrees?
5. What is the relationship between the degree of a polynomial and its graph?

**Tip:** When multiplying polynomials, it's best to expand step by step to avoid mistakes.

Find a polynomial function of degree 3 with real coefficients that has the given zeros: -1, 2, -4.

Discover how to find a polynomial function of degree 3 with real coefficients, given zeros -1, 2, and -4. This detailed guide covers essential algebra concepts, including polynomial construction from roots, with step-by-step expansion for clarity. Ideal for high school students studying polynomial functions.

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