I have a HP Pavilion Mini Desktop - 300-030 which was running Windows 10 until two days ago. I had attempted to install Linux Mint 18.3 several times in the past. It always failed at rebooting due to a Grub Error. This time, I prepared a USB Flash Drive with Linux Mint 19.1. The install was successful! But I noticed that the on-board Wi-Fi
device was not being detected or working. The device that came with the computer is a Broadcom BCM4312 802.11b/g/n. I spent several days looking for a reason or a solution. I found nothing. I remembered that I had a USB Network device that I had used on an ancient Windows Me computer. I found it and plugged it in and it was detected!! Up to that moment, My new Linux Mint 19.1 computer was convinced it was connected to my network via ethernet. I have't used a ethernet cord on this computer in many months. I went into the Control Panel and selected the Network Icon. I removed the entry that stated that the computer was using ethernet. The entry for it's new wireless connection was already present.
I restarted and expected that the change I had made would stay. It did not. It still lists both ethenet connection and WiFi Connection.
But so long as everything is working, I will accept the computer's vagueries.

Ha Ha, that's a blast from the past. I had one of those. Around 2015 the firmware was dropped from the kernel on the basis that 'Nobody uses that old s*** anymore.'
But I was using it. My BCM4312 definitely did not do 802.11n, only b, & g. Later parts may have.

Slackware do a kernel-firmware package with 'all that old s***' still catered for. Others may do as well. Mint-19.0 supplies the module, but not the firmware.