Whenever you are cooking more than one vegetable together, you want to choose vegetables that cook in the same amount of time. For that reason, carrots can be roasted with broccoli as long at they are cut into smaller pieces or sliced thin. Potatoes will need an additional 10-20 minutes to roast depending on the size. Start the potatoes in the oven first and then add the broccoli during the last 20 minutes of cooking.