diff --git a/src/polynomial/commitment.rs b/src/polynomial/commitment.rs
index 610fbc55..7f9b44bd 100644
--- a/src/polynomial/commitment.rs
+++ b/src/polynomial/commitment.rs
@@ -4,7 +4,6 @@ use crate::fri::{prover::fri_proof, verifier::verify_fri_proof, FriConfig};
 use crate::merkle_tree::MerkleTree;
 use crate::plonk_challenger::Challenger;
 use crate::plonk_common::reduce_with_powers;
-use crate::polynomial::old_polynomial::Polynomial;
 use crate::polynomial::polynomial::PolynomialCoeffs;
 use crate::proof::{FriProof, Hash};
 use crate::util::{log2_strict, reverse_index_bits_in_place, transpose};
@@ -84,8 +83,9 @@ impl<F: Field> ListPolynomialCommitment<F> {
             .polynomials
             .iter()
             .rev()
-            .map(|p| p.clone().into())
-            .fold(Polynomial::empty(), |acc, p| acc.scalar_mul(alpha).add(&p));
+            .fold(PolynomialCoeffs::zero(self.degree), |acc, p| {
+                &(&acc * alpha) + &p
+            });
         // Scale evaluations by `alpha`.
         let composition_evals = evaluations
             .iter()
@@ -118,7 +118,11 @@ impl<F: Field> ListPolynomialCommitment<F> {
 
     /// Given `points=(x_i)`, `evals=(y_i)` and `poly=P` with `P(x_i)=y_i`, computes the polynomial
     /// `Q=(P-I)/Z` where `I` interpolates `(x_i, y_i)` and `Z` is the vanishing polynomial on `(x_i)`.
-    fn compute_quotient(points: &[F], evals: &[F], poly: &Polynomial<F>) -> Polynomial<F> {
+    fn compute_quotient(
+        points: &[F],
+        evals: &[F],
+        poly: &PolynomialCoeffs<F>,
+    ) -> PolynomialCoeffs<F> {
         let pairs = points
             .iter()
             .zip(evals)
@@ -126,18 +130,15 @@ impl<F: Field> ListPolynomialCommitment<F> {
             .collect::<Vec<_>>();
         debug_assert!(pairs.iter().all(|&(x, e)| poly.eval(x) == e));
 
-        let interpolant: Polynomial<F> = interpolant(&pairs).into();
-        let denominator = points
-            .iter()
-            .fold(Polynomial::from(vec![F::ONE]), |acc, &x| {
-                acc.mul(&vec![-x, F::ONE].into())
-            });
-        let numerator = poly.add(&interpolant.neg());
-        let (mut quotient, rem) = numerator.polynomial_division(&denominator);
+        let interpolant = interpolant(&pairs);
+        let denominator = points.iter().fold(PolynomialCoeffs::one(), |acc, &x| {
+            &acc * &PolynomialCoeffs::new(vec![-x, F::ONE])
+        });
+        let numerator = poly - &interpolant;
+        let (mut quotient, rem) = numerator.div_rem(&denominator);
         debug_assert!(rem.is_zero());
 
-        quotient.pad((quotient.degree() + 1).next_power_of_two());
-        quotient
+        quotient.padded(quotient.degree_plus_one().next_power_of_two())
     }
 }
 
diff --git a/src/polynomial/polynomial.rs b/src/polynomial/polynomial.rs
index 113a6d2a..a0cdeb8b 100644
--- a/src/polynomial/polynomial.rs
+++ b/src/polynomial/polynomial.rs
@@ -1,7 +1,10 @@
+use std::cmp::max;
+use std::ops::{Add, Mul, Sub};
+
 use crate::field::fft::{fft, ifft};
 use crate::field::field::Field;
+use crate::polynomial::old_polynomial::Polynomial;
 use crate::util::log2_strict;
-use std::slice::Iter;
 
 /// A polynomial in point-value form.
 ///
@@ -65,6 +68,14 @@ impl<F: Field> PolynomialCoeffs<F> {
         Self::new(vec![F::ZERO; len])
     }
 
+    pub(crate) fn one() -> Self {
+        Self::new(vec![F::ONE])
+    }
+
+    pub(crate) fn is_zero(&self) -> bool {
+        self.coeffs.iter().all(|x| x.is_zero())
+    }
+
     /// The number of coefficients. This does not filter out any zero coefficients, so it is not
     /// necessarily related to the degree.
     pub(crate) fn len(&self) -> usize {
@@ -93,13 +104,13 @@ impl<F: Field> PolynomialCoeffs<F> {
         polys.into_iter().map(|p| p.lde(rate_bits)).collect()
     }
 
-    pub(crate) fn lde(self, rate_bits: usize) -> Self {
-        let original_size = self.len();
-        let lde_size = original_size << rate_bits;
-        let Self { mut coeffs } = self;
-        for _ in 0..(lde_size - original_size) {
-            coeffs.push(F::ZERO);
-        }
+    pub(crate) fn lde(&self, rate_bits: usize) -> Self {
+        self.padded(self.len() << rate_bits)
+    }
+
+    pub(crate) fn padded(&self, new_len: usize) -> Self {
+        let mut coeffs = self.coeffs.clone();
+        coeffs.resize(new_len, F::ZERO);
         Self { coeffs }
     }
 
@@ -109,13 +120,20 @@ impl<F: Field> PolynomialCoeffs<F> {
     }
 
     /// Degree of the polynomial + 1.
-    fn degree_plus_one(&self) -> usize {
+    pub(crate) fn degree_plus_one(&self) -> usize {
         (0usize..self.len())
             .rev()
             .find(|&i| self.coeffs[i].is_nonzero())
             .map_or(0, |i| i + 1)
     }
 
+    pub(crate) fn div_rem(&self, rhs: &Self) -> (Self, Self) {
+        let lhs = Polynomial::from(self.clone());
+        let rhs = Polynomial::from(rhs.clone());
+        let (q, r) = lhs.polynomial_long_division(&rhs);
+        (q.into(), r.into())
+    }
+
     pub fn fft(self) -> PolynomialValues<F> {
         fft(self)
     }
@@ -137,11 +155,69 @@ impl<F: Field> From<Vec<F>> for PolynomialCoeffs<F> {
     }
 }
 
+impl<F: Field> Add for &PolynomialCoeffs<F> {
+    type Output = PolynomialCoeffs<F>;
+
+    fn add(self, rhs: Self) -> Self::Output {
+        let len = max(self.len(), rhs.len());
+        let mut coeffs = self.coeffs.clone();
+        coeffs.resize(len, F::ZERO);
+        for (i, &c) in rhs.coeffs.iter().enumerate() {
+            coeffs[i] += c;
+        }
+        PolynomialCoeffs::new(coeffs)
+    }
+}
+
+impl<F: Field> Sub for &PolynomialCoeffs<F> {
+    type Output = PolynomialCoeffs<F>;
+
+    fn sub(self, rhs: Self) -> Self::Output {
+        let len = max(self.len(), rhs.len());
+        let mut coeffs = self.coeffs.clone();
+        coeffs.resize(len, F::ZERO);
+        for (i, &c) in rhs.coeffs.iter().enumerate() {
+            coeffs[i] -= c;
+        }
+        PolynomialCoeffs::new(coeffs)
+    }
+}
+
+impl<F: Field> Mul<F> for &PolynomialCoeffs<F> {
+    type Output = PolynomialCoeffs<F>;
+
+    fn mul(self, rhs: F) -> Self::Output {
+        let coeffs = self.coeffs.iter().map(|&x| rhs * x).collect();
+        PolynomialCoeffs::new(coeffs)
+    }
+}
+
+impl<F: Field> Mul for &PolynomialCoeffs<F> {
+    type Output = PolynomialCoeffs<F>;
+
+    fn mul(self, rhs: Self) -> Self::Output {
+        let new_len = (self.len() + rhs.len()).next_power_of_two();
+        let a = self.padded(new_len);
+        let b = rhs.padded(new_len);
+        let a_evals = a.fft();
+        let b_evals = b.fft();
+
+        let mul_evals: Vec<F> = a_evals
+            .values
+            .into_iter()
+            .zip(b_evals.values)
+            .map(|(pa, pb)| pa * pb)
+            .collect();
+        ifft(mul_evals.into())
+    }
+}
+
 #[cfg(test)]
 mod tests {
-    use super::*;
     use crate::field::crandall_field::CrandallField;
 
+    use super::*;
+
     #[test]
     fn test_coset_fft() {
         type F = CrandallField;