diff --git a/src/plonk/prover.rs b/src/plonk/prover.rs
index 6c57217c..be356d9f 100644
--- a/src/plonk/prover.rs
+++ b/src/plonk/prover.rs
@@ -100,7 +100,7 @@ pub(crate) fn prove<F: RichField + Extendable<D>, const D: usize>(
     let plonk_z_vecs = timed!(
         timing,
         "compute Z's",
-        compute_zs(&partial_products, common_data)
+        compute_zs(&mut partial_products, common_data)
     );
 
     // The first polynomial in `partial_products` represent the final product used in the
@@ -286,24 +286,26 @@ fn wires_permutation_partial_products<F: RichField + Extendable<D>, const D: usi
 }
 
 fn compute_zs<F: RichField + Extendable<D>, const D: usize>(
-    partial_products: &[Vec<PolynomialValues<F>>],
+    partial_products: &mut [Vec<PolynomialValues<F>>],
     common_data: &CommonCircuitData<F, D>,
 ) -> Vec<PolynomialValues<F>> {
     (0..common_data.config.num_challenges)
-        .map(|i| compute_z(&partial_products[i], common_data))
+        .map(|i| compute_z(&mut partial_products[i], common_data))
         .collect()
 }
 
 /// Compute the `Z` polynomial by reusing the computations done in `wires_permutation_partial_products`.
 fn compute_z<F: RichField + Extendable<D>, const D: usize>(
-    partial_products: &[PolynomialValues<F>],
+    partial_products: &mut [PolynomialValues<F>],
     common_data: &CommonCircuitData<F, D>,
 ) -> PolynomialValues<F> {
     let mut plonk_z_points = vec![F::ONE];
     for i in 1..common_data.degree() {
-        let quotient = partial_products[0].values[i - 1];
         let last = *plonk_z_points.last().unwrap();
-        plonk_z_points.push(last * quotient);
+        for q in partial_products.iter_mut() {
+            q.values[i - 1] *= last;
+        }
+        plonk_z_points.push(partial_products[0].values[i - 1]);
     }
     plonk_z_points.into()
 }
diff --git a/src/plonk/vanishing_poly.rs b/src/plonk/vanishing_poly.rs
index 43e52994..899d69a6 100644
--- a/src/plonk/vanishing_poly.rs
+++ b/src/plonk/vanishing_poly.rs
@@ -70,13 +70,13 @@ pub(crate) fn eval_vanishing_poly<F: RichField + Extendable<D>, const D: usize>(
             &numerator_values,
             &denominator_values,
             current_partial_products,
+            z_x,
             max_degree,
         );
         vanishing_partial_products_terms.extend(partial_product_checks);
 
         let v_shift_term = *current_partial_products.last().unwrap()
             * numerator_values[final_num_prod..].iter().copied().product()
-            * z_x
             - z_gz
                 * denominator_values[final_num_prod..]
                     .iter()
@@ -180,13 +180,13 @@ pub(crate) fn eval_vanishing_poly_base_batch<F: RichField + Extendable<D>, const
                 &numerator_values,
                 &denominator_values,
                 current_partial_products,
+                z_x,
                 max_degree,
             );
             vanishing_partial_products_terms.extend(partial_product_checks);
 
             let v_shift_term = *current_partial_products.last().unwrap()
                 * numerator_values[final_num_prod..].iter().copied().product()
-                * z_x
                 - z_gz
                     * denominator_values[final_num_prod..]
                         .iter()
@@ -376,6 +376,7 @@ pub(crate) fn eval_vanishing_poly_recursively<F: RichField + Extendable<D>, cons
             &numerator_values,
             &denominator_values,
             current_partial_products,
+            z_x,
             max_degree,
         );
         vanishing_partial_products_terms.extend(partial_product_checks);
@@ -390,7 +391,7 @@ pub(crate) fn eval_vanishing_poly_recursively<F: RichField + Extendable<D>, cons
             v.push(z_gz);
             v
         });
-        let v_shift_term = builder.mul_sub_extension(nume_acc, z_x, z_gz_denominators);
+        let v_shift_term = builder.sub_extension(nume_acc, z_gz_denominators);
         vanishing_v_shift_terms.push(v_shift_term);
     }
 
diff --git a/src/util/partial_products.rs b/src/util/partial_products.rs
index c3c4659a..92419a56 100644
--- a/src/util/partial_products.rs
+++ b/src/util/partial_products.rs
@@ -34,12 +34,12 @@ pub fn check_partial_products<F: Field>(
     numerators: &[F],
     denominators: &[F],
     partials: &[F],
+    mut acc: F,
     max_degree: usize,
 ) -> Vec<F> {
     debug_assert!(max_degree > 1);
     let mut partials = partials.iter();
     let mut res = Vec::new();
-    let mut acc = F::ONE;
     let chunk_size = max_degree;
     for (nume_chunk, deno_chunk) in numerators
         .chunks_exact(chunk_size)
@@ -60,12 +60,12 @@ pub fn check_partial_products_recursively<F: RichField + Extendable<D>, const D:
     numerators: &[ExtensionTarget<D>],
     denominators: &[ExtensionTarget<D>],
     partials: &[ExtensionTarget<D>],
+    mut acc: ExtensionTarget<D>,
     max_degree: usize,
 ) -> Vec<ExtensionTarget<D>> {
     debug_assert!(max_degree > 1);
     let mut partials = partials.iter();
     let mut res = Vec::new();
-    let mut acc = builder.one_extension();
     let chunk_size = max_degree;
     for (nume_chunk, deno_chunk) in numerators
         .chunks_exact(chunk_size)
@@ -108,7 +108,7 @@ mod tests {
 
         let nums = num_partial_products(v.len(), 2);
         assert_eq!(p.len(), nums.0);
-        assert!(check_partial_products(&v, &denominators, &p, 2)
+        assert!(check_partial_products(&v, &denominators, &p, F::ONE, 2)
             .iter()
             .all(|x| x.is_zero()));
         assert_eq!(
@@ -130,7 +130,7 @@ mod tests {
         );
         let nums = num_partial_products(v.len(), 3);
         assert_eq!(p.len(), nums.0);
-        assert!(check_partial_products(&v, &denominators, &p, 3)
+        assert!(check_partial_products(&v, &denominators, &p, F::ONE, 3)
             .iter()
             .all(|x| x.is_zero()));
         assert_eq!(