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Section 5.2 : Zeroes/Roots of Polynomials

For problems 1 – 6 list all of the zeros of the polynomial and give their multiplicities.

1. \(f\left( x \right) = {x^2} + 2x - 120\)
2. \(R\left( x \right) = {x^2} + 12x + 32\)
3. \(h\left( x \right) = 4{x^3} + {x^2} - 3x\)
4. \(A\left( x \right) = {x^5} + 2{x^4} - 35{x^3} + 92{x^2} - 92x + 32 = {\left( {x - 1} \right)^2}\left( {x + 8} \right){\left( {x - 2} \right)^2}\)
5. \(Q\left( x \right) = {x^{10}} + 17{x^9} + 115{x^8} + 387{x^7} + 648{x^6} + 432{x^5} = {x^5}{\left( {x + 3} \right)^3}{\left( {x + 4} \right)^2}\)
6. \(g\left( x \right) = {x^8} + 2{x^7} - 14{x^6} - 16{x^5} + 49{x^4} + 62{x^3} - 44{x^2} - 88x - 32 = \left( {x + 4} \right){\left( {x + 1} \right)^4}{\left( {x - 2} \right)^3}\)

For problems 7 – 11 \(x = r\) is a root of the given polynomial. Find the other two roots and write the polynomial in fully factored form.

1. \(P\left( x \right) = {x^4} - 3{x^3} - 18{x^2}\) ; \(r = 6\)
2. \(P\left( x \right) = {x^3} + {x^2} - 46x + 80\) ; \(r = - 8\)
3. \(P\left( x \right) = {x^3} - 9{x^2} + 26x - 24\) ; \(r = 3\)
4. \(P\left( x \right) = 12{x^3} + 13{x^2} - 1\) ; \(r = - 1\)
5. \(P\left( x \right) = 4{x^3} + 11{x^2} - 134x - 105\) ; \(r = 5\)

For problems 12 – 14 determine the smallest possible degree for a polynomial with the given zeros and their multiplicities.

1. \({r_1} = - 2\) (multiplicity 1), \({r_2} = 1\) (multiplicity 1), \({r_3} = 4\) (multiplicity 1)
2. \({r_1} = 3\) (multiplicity 4), \({r_2} = - 5\) (multiplicity 1)
3. \({r_1} = 7\) (multiplicity 2), \({r_2} = 4\) (multiplicity 7), \({r_3} = - 10\) (multiplicity 5)
4. A 7th degree polynomial has roots \({r_1} = - 9\) (multiplicity 2) and \({r_{\,2}} = 3\) (multiplicity 1). What is the maximum number of remaining roots for the polynomial?