To write a polynomial function of least degree with integer coefficients based on the given zeros, we follow these steps:

Given Zeros:

• $-4$
• $2$
• $\frac{5}{2}$

Step 1: Create factors corresponding to the zeros.

Each zero $x = r$ corresponds to a factor of the form $(x - r)$.

• For $x = -4$, the factor is $(x + 4)$.
• For $x = 2$, the factor is $(x - 2)$.
• For $x = \frac{5}{2}$, the factor is $(x - \frac{5}{2})$.

Step 2: Remove any fractions by multiplying the factors.

To ensure integer coefficients, we eliminate the fraction in the third factor. Multiply both sides of $(x - \frac{5}{2})$ by 2 to make the coefficient of $x$ an integer: $(x - \frac{5}{2}) = \frac{1}{2}(2x - 5).$ So, we multiply the entire polynomial by 2 later to eliminate the fraction.

Step 3: Multiply the factors.

Now we have the factors: $(x + 4), \quad (x - 2), \quad \text{and} \quad (2x - 5).$ Multiply these factors together to get the polynomial.

First, multiply the first two factors: $(x + 4)(x - 2) = x^2 - 2x + 4x - 8 = x^2 + 2x - 8.$

Now multiply this result by $(2x - 5)$: $(x^2 + 2x - 8)(2x - 5).$

Step 4: Expand the expression:

$(x^2 + 2x - 8)(2x - 5) = x^2(2x - 5) + 2x(2x - 5) - 8(2x - 5).$ Now distribute each term: $= 2x^3 - 5x^2 + 4x^2 - 10x - 16x + 40.$ Combine like terms: $= 2x^3 - x^2 - 26x + 40.$

Step 5: Final polynomial function.

The polynomial function of least degree with integer coefficients is: $2x^3 - x^2 - 26x + 40.$

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Would you like a more detailed explanation of any step or have any questions?

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Here are some related questions you might find interesting:

1. How do you find the degree of a polynomial from its factors?
2. What are the properties of polynomials with integer coefficients?
3. Can a polynomial have fractional coefficients even if the zeros are integers?
4. How would the polynomial change if the zero $\frac{5}{2}$ were a rational number with a different denominator?
5. What’s the relationship between the number of zeros and the degree of a polynomial?

Tip: When working with fractional zeros, always multiply by the denominator to remove fractions and get integer coefficients!