Yes, you have to give your dog supplements when you are home cooking. Chicken, brown rice, and carrots alone will not balance many things in the diet that the dog needs. You can create long term problems by not balancing the diet properly.
There are lots of us on YT that home cook - you should come by the diet thread and post there. I am a firm believer that if you are going to home cook, you should get a veterinary nutritionist to formulate the diet for you.
I have three dogs and home cook for all three - two are on medical diets but Teddy is on a regular adult diet that I paid $25 for on
www.petdiets.com -- run by a veterinary nutritionist so it is balanced
correctly. I also bought the "Balance It" supplement that just mixes the vitamins in and goes with the diet I purchased. The diet tells you exactly how much of each ingredient you need to mix to meet the dog's daily nutritional requirements. You can pick your main ingredients too. Teddy gets roasted chicken, brown rice, and veggies as his main items.
I've home cooked for many years and although I've done it many ways I find the balance it supplement to be the easiest. I sprinkle it on top of my food mixture and I'm done.
Good luck!