Sunday, October 27, 2019

College Algebra, Chapter 4, 4.5, Section 4.5, Problem 40

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 abi is also a zero of P. In our case, we have zeros of 3,1+i and 1i. Thus

P(x)=[x(3)][x(1+i)][x(1i)]ModelP(x)=(x+3)[(x1)i][(x1)+i]RegroupP(x)=(x+3)[(x1)2i2]Difference of groupP(x)=(x+3)[x22x+1+1]Expand, recall that i2=1P(x)=(x+3)(x22x+2)SimplifyP(x)=x32x2+2x+3x26x+6ExpandP(x)=x3+x24x+6Combine like terms

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