Sunday, December 30, 2018

College Algebra, Chapter 4, 4.5, Section 4.5, Problem 44

Find a polynomial P(x) that has degree 5 with integer coefficient and zeros 12,1 and i, and leading coefficient 4; the zero 1 has multiplicity 2.
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 12,1,i and i. Thus the required polynomial has the form.

P(x)=a(x12)[x(1)]2(xi)[x(i)]ModelP(x)=a(x12)(x+1)2(xi)(x+i)SimplifyP(x)=a(x12)(x+1)2(x2i2)Difference of squaresP(x)=a(x12)(x+1)2(x2+1)Recall that i2=1 P(x)=a(x+1)2[x3+x12x212]Apply FOIL methodP(x)=a[x2+2x+1][x3+x12x212]ExpandP(x)=a[x5+x312x412x2+2x4+2x2x3x+x3+x12x212]ExpandP(x)=a[x5+32x4+x3+x212]Simplify and combine like termsP(x)=4[x5+32x4+x3+x212]Substitute a=4 to be the leading coefficientP(x)=4x5+6x4+4x3+4x22

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