Being able to use algebra is essential to studying physics. Being good with algebra can make studying physics easier than it would be if you were not good with algebra. However, being good with algebra will not ensure that you will find studying physics to be easy.
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After my opinion it is not true: if you are good at mathematics you must be good at chemistry and physics.
That should probably be easy. Try it out to be sure.
Calculus will help but there is more to physics than just that.
No, a person who is good at math will have the best chance of doing well in physics.
There is no way to know. Some aspects of calculus depend on a good understanding of algebra, others don't. It all depends on which aspects of algebra you were good at and also whether or not you retain that edge.