Question

Find o polynomial function P(x) such that P has the degree and P(x)=0 has the root (s) listed. Degree of P=3 and zeros: -5 and 4-8i.

Answer 1

To construct polynomial P(x) with degree 3 and zeros at -5 and 4-8i, we multiply the factors (x + 5), (x - (4 - 8i)), and (x - (4 + 8i)) together. The resulting polynomial, after expanding and simplifying, is P(x) = x^3 - 3x^2 - 8x + 240.

Step-by-step explanation:

To find a polynomial function P(x) with a degree of 3 and given zeros -5 and 4-8i, we need to use the fact that complex zeros come in conjugate pairs for polynomials with real coefficients. This means that if 4-8i is a zero, its conjugate 4+8i is also a zero. A polynomial with these zeros can be constructed by multiplying the factors associated with each zero together.

We begin with the factors for the zeros:

• (x + 5) for the zero at x = -5
• (x - (4 - 8i)) for the zero at x = 4 - 8i
• (x - (4 + 8i)) for the zero at x = 4 + 8i

The polynomial P(x) is then the product of these factors:

P(x) = (x + 5)(x - (4 - 8i))(x - (4 + 8i))

To find the expanded form, we first multiply the factors for the complex zeros:

(x - (4 - 8i))(x - (4 + 8i)) = x2 - (4 + 8i)x - (4 - 8i)x + (4 - 8i)(4 + 8i)

(4 - 8i)(4 + 8i) is 16 + 64, since (-8i)(8i) is 64i2, and i2 = -1. Therefore, the product of the complex conjugates is 16 - 64, which equals -48.

Now we simplify and combine like terms to get:

x2 - 8x + 48

Then we multiply this with the remaining factor:

P(x) = (x + 5)(x2 - 8x + 48)

Expand and simplify to get the final polynomial:

P(x) = x3 - 3x2 - 8x + 240

Answer 2

A polynomial function P(x) with a degree of 3 and the given roots -5, 4 - 8i is:

`\[ P(x) = (x + 5)(x - 4 + 8i)(x - 4 - 8i) \]`

Step-by-step explanation:

To construct a polynomial function with the specified roots, we use the factored form of a polynomial. The degree of the polynomial is 3, and the roots are -5, 4 - 8i, and 4 + 8i.

The factorization involves expressions of the form (x - r), where r is a root. In this case, the factorization is:

`\[ P(x) = (x + 5)(x - (4 - 8i))(x - (4 + 8i)) \]`

This expression represents a cubic polynomial with the given roots. It follows the fundamental theorem of algebra, which states that a polynomial of degree
`\( n \)` will have exactly
`\( n \)` roots.

In summary, the final answer provides the polynomial function
`\( P(x) \)` in factored form, considering the specified roots and the given degree of 3. The factors include
`\((x + 5)\)` for the root -5, and
`\((x - (4 - 8i))(x - (4 + 8i))\)`for the complex conjugate roots 4 - 8i and 4 + 8i.