(x-3)(x-i)(x-(2+i))

Jasonbradley

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Jun 17, 2024

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Finding the Polynomial from its Roots

Given the roots of a polynomial, we can construct the polynomial itself. This is based on the Factor Theorem, which states that a polynomial has a factor (x - a) if and only if 'a' is a root of the polynomial.

We are given the roots:

• x = 3
• x = i
• x = 2 + i

From these roots, we can construct the factors:

• (x - 3)
• (x - i)
• (x - (2 + i))

Multiplying these factors together will give us the polynomial:

(x - 3)(x - i)(x - (2 + i))

To find the polynomial, we need to expand this expression:

Step 1: Expand the first two factors.

(x - 3)(x - i) = x² - ix - 3x + 3i = x² - (3 + i)x + 3i

Step 2: Multiply the result from Step 1 with the third factor.

(x² - (3 + i)x + 3i)(x - (2 + i)) = x³ - (2 + i)x² - (3 + i)x² + (3 + i)(2 + i)x + 3ix - 3i(2 + i)

Step 3: Simplify the expression.

x³ - (2 + i + 3 + i)x² + (6 + 3i + 2i - i²)x + 3ix - 6i - 3i²

Step 4: Remember that i² = -1.

x³ - (5 + 2i)x² + (8 + 5i)x + 3ix + 6 - 3

Step 5: Combine like terms.

x³ - (5 + 2i)x² + (8 + 8i)x + 6

Therefore, the polynomial with the given roots is x³ - (5 + 2i)x² + (8 + 8i)x + 6.

Note: This polynomial has complex coefficients due to the complex roots.