Question

f(x) = x^5 - 8x^4 + 21x^3 - 12x^2 - 22x + 20

three roots of this polynomial function are -1, 1, and 3 + i.

which of the following describes the number and nature of all the roots of this function?

• f(x) has two real roots and one imaginary root.
• f(x) has three real roots.
• f(x) has five real roots.
• f(x) has three real roots and two imaginary roots.

Explanation:

🆕 New Concept Discovered: Complex Conjugate Root Theorem
Complex roots of real polynomials always come in conjugate pairs.

Step 1: Identify total number of roots

The polynomial function is given by:

$$ f(x) = x^5 - 8x^4 + 21x^3 - 12x^2 - 22x + 20 $$

The degree of the polynomial is \( 5 \) (the highest exponent of \( x \)). By the Fundamental Theorem of Algebra, a polynomial of degree \( n \) has exactly \( n \) complex roots (including real and imaginary roots, counting multiplicities).

Therefore, \( f(x) \) has exactly \( 5 \) roots in total.

Step 2: Apply the Complex Conjugate Root Theorem

We are given three of the roots:

$$ x_1 = -1 $$

$$ x_2 = 1 $$

$$ x_3 = 3 + i $$

Since the coefficients of the polynomial \( f(x) \) are all real numbers, any complex (imaginary) roots must occur in conjugate pairs.

Because \( 3 + i \) is a root, its complex conjugate, \( 3 - i \), must also be a root of the function:

$$ x_4 = 3 - i $$

This gives us a total of four identified roots:

• Real roots: \( -1 \), \( 1 \)
• Imaginary roots: \( 3 + i \), \( 3 - i \)

Step 3: Determine the final root

Since the polynomial has a degree of \( 5 \), there is one remaining root \( x_5 \) to find.

Because imaginary roots can only exist in pairs for polynomials with real coefficients, the fifth root cannot be imaginary (otherwise, it would lack a conjugate partner). Therefore, the fifth root must be a real root.

Counting all the roots:

• Real roots: \( 3 \) (the two known real roots \( -1 \) and \( 1 \), plus the fifth remaining root)
• Imaginary roots: \( 2 \) (the conjugate pair \( 3 + i \) and \( 3 - i \))

Answer:

\( f(x) \) has three real roots and two imaginary roots.