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If you are looking for some problems with solutions you can find some by clicking on the "Practice Problems" link above.
If you are looking for some problems with solutions you can find some by clicking on the "Practice Problems" link above.
Section 5.2 : Zeroes/Roots of Polynomials
For problems 1 – 6 list all of the zeros of the polynomial and give their multiplicities.
- \(f\left( x \right) = {x^2} + 2x - 120\)
- \(R\left( x \right) = {x^2} + 12x + 32\)
- \(h\left( x \right) = 4{x^3} + {x^2} - 3x\)
- \(A\left( x \right) = {x^5} + 2{x^4} - 35{x^3} + 92{x^2} - 92x + 32 = {\left( {x - 1} \right)^2}\left( {x + 8} \right){\left( {x - 2} \right)^2}\)
- \(Q\left( x \right) = {x^{10}} + 17{x^9} + 115{x^8} + 387{x^7} + 648{x^6} + 432{x^5} = {x^5}{\left( {x + 3} \right)^3}{\left( {x + 4} \right)^2}\)
- \(g\left( x \right) = {x^8} + 2{x^7} - 14{x^6} - 16{x^5} + 49{x^4} + 62{x^3} - 44{x^2} - 88x - 32 = \left( {x + 4} \right){\left( {x + 1} \right)^4}{\left( {x - 2} \right)^3}\)
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.
- \(P\left( x \right) = {x^4} - 3{x^3} - 18{x^2}\) ; \(r = 6\)
- \(P\left( x \right) = {x^3} + {x^2} - 46x + 80\) ; \(r = - 8\)
- \(P\left( x \right) = {x^3} - 9{x^2} + 26x - 24\) ; \(r = 3\)
- \(P\left( x \right) = 12{x^3} + 13{x^2} - 1\) ; \(r = - 1\)
- \(P\left( x \right) = 4{x^3} + 11{x^2} - 134x - 105\) ; \(r = 5\)
For problems 12 – 14 determine the smallest possible degree for a polynomial with the given zeros and their multiplicities.
- \({r_1} = - 2\) (multiplicity 1), \({r_2} = 1\) (multiplicity 1), \({r_3} = 4\) (multiplicity 1)
- \({r_1} = 3\) (multiplicity 4), \({r_2} = - 5\) (multiplicity 1)
- \({r_1} = 7\) (multiplicity 2), \({r_2} = 4\) (multiplicity 7), \({r_3} = - 10\) (multiplicity 5)
- 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?