diff --git a/doc/whats-new.rst b/doc/whats-new.rst
index 17af655c02e..9b40a323f39 100644
--- a/doc/whats-new.rst
+++ b/doc/whats-new.rst
@@ -71,6 +71,8 @@ Bug fixes
   By `Kai Mühlbauer <https://github.com/kmuehlbauer>`_.
 - Use zarr-fixture to prevent thread leakage errors (:pull:`9967`).
   By `Kai Mühlbauer <https://github.com/kmuehlbauer>`_.
+- Fix weighted ``polyfit`` for arrays with more than two dimensions (:issue:`9972`, :pull:`9974`).
+  By `Mattia Almansi <https://github.com/malmans2>`_.
 
 Documentation
 ~~~~~~~~~~~~~
diff --git a/xarray/core/dataset.py b/xarray/core/dataset.py
index a943d9bfc57..74f90ce9eea 100644
--- a/xarray/core/dataset.py
+++ b/xarray/core/dataset.py
@@ -9206,7 +9206,7 @@ def polyfit(
 
             present_dims.update(other_dims)
             if w is not None:
-                rhs = rhs * w[:, np.newaxis]
+                rhs = rhs * w.reshape(-1, *((1,) * len(other_dims)))
 
             with warnings.catch_warnings():
                 if full:  # Copy np.polyfit behavior
diff --git a/xarray/tests/test_dataset.py b/xarray/tests/test_dataset.py
index 8a90a05a4e3..f3867bd67d2 100644
--- a/xarray/tests/test_dataset.py
+++ b/xarray/tests/test_dataset.py
@@ -6685,11 +6685,15 @@ def test_polyfit_output(self) -> None:
         assert len(out.data_vars) == 0
 
     def test_polyfit_weighted(self) -> None:
-        # Make sure weighted polyfit does not change the original object (issue #5644)
         ds = create_test_data(seed=1)
+        ds = ds.broadcast_like(ds)  # test more than 2 dimensions (issue #9972)
         ds_copy = ds.copy(deep=True)
 
-        ds.polyfit("dim2", 2, w=np.arange(ds.sizes["dim2"]))
+        expected = ds.polyfit("dim2", 2)
+        actual = ds.polyfit("dim2", 2, w=np.ones(ds.sizes["dim2"]))
+        xr.testing.assert_identical(expected, actual)
+
+        # Make sure weighted polyfit does not change the original object (issue #5644)
         xr.testing.assert_identical(ds, ds_copy)
 
     def test_polyfit_coord(self) -> None:
diff --git a/xarray/tests/test_datatree.py b/xarray/tests/test_datatree.py
index 7b295128d63..21870050034 100644
--- a/xarray/tests/test_datatree.py
+++ b/xarray/tests/test_datatree.py
@@ -1240,8 +1240,12 @@ def test_repr_inherited_dims(self) -> None:
     )
     def test_doc_example(self) -> None:
         # regression test for https://github.com/pydata/xarray/issues/9499
-        time = xr.DataArray(data=["2022-01", "2023-01"], dims="time")
-        stations = xr.DataArray(data=list("abcdef"), dims="station")
+        time = xr.DataArray(
+            data=np.array(["2022-01", "2023-01"], dtype="<U7"), dims="time"
+        )
+        stations = xr.DataArray(
+            data=np.array(list("abcdef"), dtype="<U1"), dims="station"
+        )
         lon = [-100, -80, -60]
         lat = [10, 20, 30]
         # Set up fake data