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Section 5.2 : Zeroes/Roots of Polynomials

For problems 1 – 3 list all of the zeros of the polynomial and give their multiplicities.

1. \(f\left( x \right) = 2{x^2} + 13x - 7\) Solution
2. \(g\left( x \right) = {x^6} - 3{x^5} - 6{x^4} + 10{x^3} + 21{x^2} + 9x = x{\left( {x - 3} \right)^2}{\left( {x + 1} \right)^3}\) Solution
3. \(\begin{align*}A\left( x \right) & = {x^8} + 2{x^7} - 29{x^6} - 76{x^5} + 199{x^4} + 722{x^3} + 261{x^2} - 648x - 432\\ & = {\left( {x + 1} \right)^2}{\left( {x - 4} \right)^2}\left( {x - 1} \right){\left( {x + 3} \right)^3}\end{align*}\) Solution

For problems 4 – 6 \(x = r\) is a root of the given polynomial. Find the other two roots and write the polynomial in fully factored form.

1. \(P\left( x \right) = {x^3} - 6{x^2} - 16x\) ; \(r = - 2\) Solution
2. \(P\left( x \right) = {x^3} - 7{x^2} - 6x + 72\) ; \(r = 4\) Solution
3. \(P\left( x \right) = 3{x^3} + 16{x^2} - 33x + 14\) ; \(r = - 7\) Solution