Find a polynomial P(x) of degree 3 with interger coefficients and zeros −3 and 1+i.
Recall that if the polynomial function P has real coefficient and if a a+bi is a zero of P, then a−bi is also a zero of P. In our case, we have zeros of −3,1+i and 1−i. Thus
P(x)=[x−(−3)][x−(1+i)][x−(1−i)]ModelP(x)=(x+3)[(x−1)−i][(x−1)+i]RegroupP(x)=(x+3)[(x−1)2−i2]Difference of groupP(x)=(x+3)[x2−2x+1+1]Expand, recall that i2=−1P(x)=(x+3)(x2−2x+2)SimplifyP(x)=x3−2x2+2x+3x2−6x+6ExpandP(x)=x3+x2−4x+6Combine like terms
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