Find a polynomial function P of the lowest possible degree, having real coefficients, a leading coefficient of 1 , and with the given zeros. \( 2+3 i,-1 \), and 2 The polynomial of lowest degree is \( \mathrm{P}(\mathrm{x})=\square \).

To find a polynomial function \( P(x) \) of the lowest possible degree with real coefficients, a leading coefficient of 1, and the given zeros \( 2+3i, -1, \) and 2, we can use the fact that complex zeros come in conjugate pairs for polynomials with real coefficients. Given zeros: \( 2+3i, -1, \) and 2 Since \( 2+3i \) is a complex zero, its conjugate \( 2-3i \) is also a zero. Therefore, the zeros of the polynomial are \( 2+3i, 2-3i, -1, \) and 2. The polynomial function \( P(x) \) can be written in factored form as: \[ P(x) = (x - (2+3i))(x - (2-3i))(x + 1)(x - 2) \] Expanding this expression will give us the polynomial function \( P(x) \) of the lowest possible degree with real coefficients and the given zeros. Simplify the expression by following steps: - step0: Calculate: \(\left(x-\left(2+3i\right)\right)\left(x-\left(2-3i\right)\right)\left(x+1\right)\left(x-2\right)\) - step1: Remove the parentheses: \(\left(x-2-3i\right)\left(x-\left(2-3i\right)\right)\left(x+1\right)\left(x-2\right)\) - step2: Remove the parentheses: \(\left(x-2-3i\right)\left(x-2+3i\right)\left(x+1\right)\left(x-2\right)\) - step3: Multiply the terms: \(\left(x^{2}-4x+13\right)\left(x+1\right)\left(x-2\right)\) - step4: Multiply the terms: \(\left(x^{3}-3x^{2}+9x+13\right)\left(x-2\right)\) - step5: Apply the distributive property: \(x^{3}\times x-x^{3}\times 2-3x^{2}\times x-\left(-3x^{2}\times 2\right)+9x\times x-9x\times 2+13x-13\times 2\) - step6: Multiply the terms: \(x^{4}-2x^{3}-3x^{3}-\left(-6x^{2}\right)+9x^{2}-18x+13x-26\) - step7: Remove the parentheses: \(x^{4}-2x^{3}-3x^{3}+6x^{2}+9x^{2}-18x+13x-26\) - step8: Subtract the terms: \(x^{4}-5x^{3}+15x^{2}-5x-26\) The polynomial function \( P(x) \) of the lowest possible degree with real coefficients, a leading coefficient of 1, and the given zeros \( 2+3i, -1, \) and 2 is: \[ P(x) = x^4 - 5x^3 + 15x^2 - 5x - 26 \]