diff --git a/src/c_kzg.h b/src/c_kzg.h
index 0f28592..51d4771 100644
--- a/src/c_kzg.h
+++ b/src/c_kzg.h
@@ -18,9 +18,9 @@
 #define C_KZG_H
 
 typedef enum {
-    C_KZG_OK = 0,
-    C_KZG_BADARGS,
-    C_KZG_ERROR,
+    C_KZG_OK = 0,  // Success!
+    C_KZG_BADARGS, // The supplied data is invalid in some way
+    C_KZG_ERROR,   // Internal error - should never occur
 } C_KZG_RET;
 
 #include <stdbool.h>
diff --git a/src/kzg_proofs.c b/src/kzg_proofs.c
index 3f5ba88..6b4de1c 100644
--- a/src/kzg_proofs.c
+++ b/src/kzg_proofs.c
@@ -52,7 +52,6 @@ bool check_proof_single(const KZGSettings *ks, const blst_p1 *commitment, const
 C_KZG_RET compute_proof_multi(blst_p1 *out, const KZGSettings *ks, poly *p, const blst_fr *x0, uint64_t n) {
     poly divisor, q;
     blst_fr x_pow_n;
-    C_KZG_RET ret;
 
     // Construct x^n - x0^n = (x - w^0)(x - w^1)...(x - w^(n-1))
     init_poly(&divisor, n + 1);
@@ -70,10 +69,7 @@ C_KZG_RET compute_proof_multi(blst_p1 *out, const KZGSettings *ks, poly *p, cons
     divisor.coeffs[n] = fr_one;
 
     // Calculate q = p / (x^n - x0^n)
-    init_poly(&q, poly_quotient_length(p, &divisor));
-    if ((ret = poly_long_div(&q, p, &divisor) != C_KZG_OK)) {
-        return C_KZG_ERROR;
-    }
+    ASSERT(poly_long_div(&q, p, &divisor) == C_KZG_OK, C_KZG_ERROR);
 
     commit_to_poly(out, ks, &q);
 
@@ -121,4 +117,4 @@ bool check_proof_multi(const KZGSettings *ks, const blst_p1 *commitment, const b
     free_poly(&interp);
 
     return pairings_verify(&commit_minus_interp, blst_p2_generator(), proof, &xn_minus_yn);
-}
\ No newline at end of file
+}
diff --git a/src/poly.c b/src/poly.c
index a4fb923..a82822d 100644
--- a/src/poly.c
+++ b/src/poly.c
@@ -24,6 +24,7 @@ static void poly_factor_div(blst_fr *out, const blst_fr *a, const blst_fr *b) {
 void init_poly(poly *out, const uint64_t length) {
     out->length = length;
     out->coeffs = length > 0 ? malloc(length * sizeof(blst_fr)): NULL;
+    // TODO: check malloc return and handle accordingly
 }
 
 void free_poly(poly *p) {
@@ -62,18 +63,22 @@ uint64_t poly_quotient_length(const poly *dividend, const poly *divisor) {
     return dividend->length >= divisor->length ? dividend->length - divisor->length + 1 : 0;
 }
 
-// `out` must have been pre-allocated to the correct size, see `poly_quotient_length()`
+// `out` must be an uninitialised poly and has space allocated for it here, which
+// must be freed by calling `free_poly()` later.
 C_KZG_RET poly_long_div(poly *out, const poly *dividend, const poly *divisor) {
     uint64_t a_pos = dividend->length - 1;
     uint64_t b_pos = divisor->length - 1;
     uint64_t diff = a_pos - b_pos;
     blst_fr a[dividend->length];
 
+    // Dividing by zero is undefined
     ASSERT(divisor->length > 0, C_KZG_BADARGS);
-    ASSERT(out->length == poly_quotient_length(dividend, divisor), C_KZG_BADARGS);
+
+    // Initialise the output polynomial
+    init_poly(out, poly_quotient_length(dividend, divisor));
 
     // If the divisor is larger than the dividend, the result is zero-length
-    if (divisor->length > dividend->length) return C_KZG_OK;
+    if (out->length == 0) return C_KZG_OK;
 
     for (uint64_t i = 0; i < dividend->length; i++) {
         a[i] = dividend->coeffs[i];
diff --git a/src/poly_test.c b/src/poly_test.c
index b5e9f0d..94bfb5c 100644
--- a/src/poly_test.c
+++ b/src/poly_test.c
@@ -33,7 +33,7 @@ void poly_div_length(void) {
 }
 
 void poly_div_0(void) {
-    blst_fr a[3], b[2], c[2], expected[2];
+    blst_fr a[3], b[2], expected[2];
     poly dividend, divisor, actual;
 
     // Calculate (x^2 - 1) / (x + 1) = x - 1
@@ -57,16 +57,15 @@ void poly_div_0(void) {
     fr_negate(&expected[0], &expected[0]);
     fr_from_uint64(&expected[1], 1);
 
-    actual.length = 2;
-    actual.coeffs = c;
-
     TEST_CHECK(C_KZG_OK == poly_long_div(&actual, &dividend, &divisor));
     TEST_CHECK(fr_equal(&expected[0], &actual.coeffs[0]));
     TEST_CHECK(fr_equal(&expected[1], &actual.coeffs[1]));
+
+    free_poly(&actual);
 }
 
 void poly_div_1(void) {
-    blst_fr a[4], b[2], c[3], expected[3];
+    blst_fr a[4], b[2], expected[3];
     poly dividend, divisor, actual;
 
     // Calculate (12x^3 - 11x^2 + 9x + 18) / (4x + 3) = 3x^2 - 5x + 6
@@ -92,13 +91,12 @@ void poly_div_1(void) {
     fr_negate(&expected[1], &expected[1]);
     fr_from_uint64(&expected[2], 3);
 
-    actual.length = 3;
-    actual.coeffs = c;
-
     TEST_CHECK(C_KZG_OK == poly_long_div(&actual, &dividend, &divisor));
     TEST_CHECK(fr_equal(&expected[0], &actual.coeffs[0]));
     TEST_CHECK(fr_equal(&expected[1], &actual.coeffs[1]));
     TEST_CHECK(fr_equal(&expected[2], &actual.coeffs[2]));
+
+    free_poly(&actual);
 }
 
 void poly_div_2(void) {
@@ -121,8 +119,6 @@ void poly_div_2(void) {
     divisor.length = 3;
     divisor.coeffs = b;
 
-    init_poly(&actual, poly_quotient_length(&dividend, &divisor));
-
     TEST_CHECK(C_KZG_OK == poly_long_div(&actual, &dividend, &divisor));
     TEST_CHECK(NULL == actual.coeffs);
 
@@ -131,7 +127,7 @@ void poly_div_2(void) {
 
 void poly_div_by_zero(void) {
     blst_fr a[2];
-    poly dividend, divisor;
+    poly dividend, divisor, dummy;
 
     // Calculate (x + 1) / 0 = FAIL
 
@@ -144,7 +140,7 @@ void poly_div_by_zero(void) {
     // Divisor
     init_poly(&divisor, 0);
 
-    TEST_CHECK(C_KZG_BADARGS == poly_long_div(NULL, &dividend, &divisor));
+    TEST_CHECK(C_KZG_BADARGS == poly_long_div(&dummy, &dividend, &divisor));
 }
 
 void poly_eval_check(void) {