Questions: Find a degree 3 polynomial with real coefficients having zeros 1 and 3-2i and a lead coefficient of 1. Write P in expanded form. Be sure to write the full equation, including P(x)=.

Find a degree 3 polynomial with real coefficients having zeros 1 and 3-2i and a lead coefficient of 1. Write P in expanded form. Be sure to write the full equation, including P(x)=.

Transcript text: Find a degree 3 polynomial with real coefficients having zeros 1 and $3-2 i$ and a lead coefficient of 1 . Write $P$ in expanded form. Be sure to write the full equation, including $P(x)=$. $\square$

Solution

Solution Steps

Step 1: Acknowledge Complex Conjugate Zeros and Real Zero

Given a real zero $r = 1$ and complex conjugate zeros $3 \pm 2i$, we know the polynomial must have real coefficients and these zeros.

Step 2: Construct the Polynomial from Its Zeros

The polynomial can be constructed as $P(x) = (x - 1)(x^2 - 2_3_x + (3^2 + 2^2))$.

Step 3: Expand the Product

Expanding the product, we get the polynomial in its expanded form: $$P(x) = x^3 -7x^2 + 19x -13$$

Final Answer:

The degree 3 polynomial with real coefficients, having at least one real zero and a pair of complex conjugate zeros, and a lead coefficient of 1 is: $$P(x) = x^3 -7x^2 + 19x -13$$

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