Unwanted high-power and frequency-agile radio-frequency (RF) signals cause saturation and nonlinear distortions in sensitive RF receivers. RF notch filters with reconfigurable responses over multiple frequency bands are highly sought after to prevent these detrimental effects. Microwave photonic (MWP) notch filters have shown increased frequency agility compared to their electronic counterparts. However, demonstrated filter schemes focus only on single notch responses, leaving them unable to simultaneously attenuate multiple interferers over a wide frequency range. In this work, we demonstrate a high-performance chip-based MWP notch filter with three independent notches widely tunable over 20 GHz. The filter exhibits low RF passband losses of 8 dB and peak notch depth greater than 40 dB with 500 MHz spectral resolution. Specifically, each notch is formed through cascaded optical processing from on-chip, low-loss Si<inline-formula><tex-math notation="LaTeX">$_3$</tex-math></inline-formula>N<inline-formula><tex-math notation="LaTeX">$_4$</tex-math></inline-formula> micro-resonators and Brillouin gain on an As<inline-formula><tex-math notation="LaTeX">$_2$</tex-math></inline-formula>S<inline-formula><tex-math notation="LaTeX">$_3$</tex-math></inline-formula> photonic chip. We use this filter to demonstrate high-performance analog RF filtering by substantially attenuating multiple interferers, and show effective image rejection during RF down-conversion in a communications receiver to enable a large reduction of error vector magnitude (EVM) from 60% to 15%. Finally, we provide performance analysis and design perspectives for future photonic integration of the proposed filter subsystem.