To solve this problem, let’s analyze the given dataTo solve this problem, let’s analyze the given

1. Degree: 4, meaning the polynomial $f(x)$ is of degree 4.
2. Zeros:
   • $-5 + 2i$ is a zero.
   • Since the polynomial has real coefficients, the complex conjugate $-5 - 2i$ must also be a zero.
   • $-1$ with multiplicity 2 is another zero.

Step 1: Write the factors corresponding to each zero

For each zero $c$, a corresponding factor is $(x - c)$.

1. For $-5 + 2i$: the factor is $(x + 5 - 2i)$.
2. For $-5 - 2i$: the factor is $(x + 5 + 2i)$.
3. For $-1$ with multiplicity 2: the factor is $(x + 1)^2$.

Step 2: Form the polynomial $f(x)$

The polynomial can be represented as: $f(x) = a \cdot (x + 5 - 2i)(x + 5 + 2i)(x + 1)^2$ where $a$ is the leading coefficient.

Step 3: Simplify the complex factors

Use the difference of squares to simplify $(x + 5 - 2i)(x + 5 + 2i)$: $(x + 5 - 2i)(x + 5 + 2i) = (x + 5)^2 - (2i)^2$ Since $(2i)^2 = -4$, this simplifies to: $(x + 5)^2 + 4$

Expanding $(x + 5)^2 + 4$: $(x + 5)^2 + 4 = x^2 + 10x + 25 + 4 = x^2 + 10x + 29$

Step 4: Substitute and expand the polynomial

Now we have: $f(x) = a \cdot (x^2 + 10x + 29)(x + 1)^2$

Expanding $(x + 1)^2$: $(x + 1)^2 = x^2 + 2x + 1$

Finally, multiply these expressions: $f(x) = a \cdot (x^2 + 10x + 29)(x^2 + 2x + 1)$

Expanding this product: $f(x) = a \cdot (x^4 + 2x^3 + x^2 + 10x^3 + 20x^2 + 10x + 29x^2 + 58x + 29)$

Combine like terms: $f(x) = a \cdot (x^4 + 12x^3 + 50x^2 + 68x + 29)$

Thus, the polynomial $f(x)$ is: $f(x) = a(x^4 + 12x^3 + 50x^2 + 68x + 29)$

This is the polynomial in terms of the variable $a$ as the leading coefficient.

Would you like further clarification or details on any part of this solution?

Related Questions

1. What is the difference between real and complex roots in polynomials?
2. How do you determine the degree of a polynomial given its roots?
3. How can you find the multiplicity of a root?
4. What happens to a polynomial if a root has a higher multiplicity?
5. How does the leading coefficient $a$ affect the shape of the polynomial graph?

Tip

When a polynomial has real coefficients, any complex roots must occur in conjugate pairs.