Find a degree 3 polynomial with real coefficients having zeros 1 and 4 i and a lead coefficient of 1. Write P in expanded form

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alenahelenash · Accepted Answer

To find a degree 3 polynomial with real coefficients that has zeros at 1 and 4i, we know that complex zeros occur in conjugate pairs. Since 4i is a zero, its conjugate -4i must also be a zero.The polynomial, let&#039;s call it P(x), can be expressed using its zeros as follows:P(x)=(x−1)(x−4i)(x+4i)Now, let&#039;s expand the expression to get the polynomial in its expanded form:P(x)=(x−1)(x2+4ix−4ix−16i2)Remember that i2=−1:P(x)=(x−1)(x2−16i2)Simplifying further:P(x)=(x−1)(x2+16)Expanding the multiplication:P(x)=x3+16x−x2−16Rearranging the terms in descending order of powers:P(x)=x3−x2+16x−16Therefore, the degree 3 polynomial with real coefficients and zeros 1 and 4i, with a lead coefficient of 1, is given by:P(x)=x3−x2+16x−16

Find a degree 3 polynomial with real coefficients

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