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Find the polynomial with rational c...

Find the polynomial with rational coefficients and whose roots are
`1 +- 2i ,4,2`

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To find the polynomial with rational coefficients whose roots are \(1 \pm 2i\), \(4\), and \(2\), we can follow these steps: ### Step 1: Identify the Roots The roots given are: - \( \alpha = 1 + 2i \) - \( \beta = 1 - 2i \) - \( \gamma = 4 \) - \( \delta = 2 \) ### Step 2: Form the Quadratic Polynomial from Complex Roots The roots \( \alpha \) and \( \beta \) can be used to form a quadratic polynomial. The sum and product of the roots can be calculated as follows: **Sum of roots:** \[ \alpha + \beta = (1 + 2i) + (1 - 2i) = 2 \] **Product of roots:** \[ \alpha \cdot \beta = (1 + 2i)(1 - 2i) = 1^2 - (2i)^2 = 1 - (-4) = 1 + 4 = 5 \] Thus, the quadratic polynomial with roots \( \alpha \) and \( \beta \) is: \[ x^2 - (\alpha + \beta)x + \alpha \cdot \beta = x^2 - 2x + 5 \] ### Step 3: Form the Quadratic Polynomial from Real Roots Next, we form a quadratic polynomial from the real roots \( \gamma \) and \( \delta \): **Sum of roots:** \[ \gamma + \delta = 4 + 2 = 6 \] **Product of roots:** \[ \gamma \cdot \delta = 4 \cdot 2 = 8 \] Thus, the quadratic polynomial with roots \( \gamma \) and \( \delta \) is: \[ x^2 - (\gamma + \delta)x + \gamma \cdot \delta = x^2 - 6x + 8 \] ### Step 4: Multiply the Two Quadratic Polynomials Now, we need to multiply the two quadratic polynomials obtained: 1. \( x^2 - 2x + 5 \) 2. \( x^2 - 6x + 8 \) The multiplication is done as follows: \[ (x^2 - 2x + 5)(x^2 - 6x + 8) \] ### Step 5: Perform the Multiplication Using the distributive property: \[ = x^2(x^2 - 6x + 8) - 2x(x^2 - 6x + 8) + 5(x^2 - 6x + 8) \] Calculating each term: 1. \( x^2(x^2 - 6x + 8) = x^4 - 6x^3 + 8x^2 \) 2. \( -2x(x^2 - 6x + 8) = -2x^3 + 12x^2 - 16x \) 3. \( 5(x^2 - 6x + 8) = 5x^2 - 30x + 40 \) Now, combine all these: \[ x^4 - 6x^3 + 8x^2 - 2x^3 + 12x^2 - 16x + 5x^2 - 30x + 40 \] ### Step 6: Combine Like Terms Combine the coefficients of like terms: - \( x^4 \): \( 1 \) - \( x^3 \): \( -6 - 2 = -8 \) - \( x^2 \): \( 8 + 12 + 5 = 25 \) - \( x \): \( -16 - 30 = -46 \) - Constant: \( 40 \) Thus, the polynomial is: \[ x^4 - 8x^3 + 25x^2 - 46x + 40 \] ### Final Answer The required polynomial is: \[ \boxed{x^4 - 8x^3 + 25x^2 - 46x + 40} \]
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