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christinewilliams
Identify the roots of -3x^3-21x^2+72x+540=0State the multiplicity of each root.
The root -5 has a multiplicity of 1, and the root 6 has a multiplicity of 2. The root 5 has a multiplicity of 1, and the root -6 has a multiplicity of 2. it could be either one of those and i dont know which one it is i think it is the first one
6 is not a root, if you put it in the equation, you do not get 0. -6 is a root, as is 5. Start with 5 as a root, you then know that you could write the eq. as\[(x-5)(2nd \deg polynomial)\]You could find out about the 2nd degree polynomial by doing a long division, or a synthetic division, which is very fast. It gives:\[(x-5)(-3x^2-36x-108)=0\]Factoring out a -3 helps to find the other (double) solution -6. If you need help with long division or synthetic division, just yell ;)
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The last step:\[(x-5)(-3x^2-36x-108)=0\]Factor out -3:\[-3(x-5)(x^2-12x-36)=0\]Factorize the 2nd degree part:\[-3(x-5)(x-6)^2=0\]Now you can see that 5 is a single root and 6 a double one.