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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?