Take the three <rank-one quadratic form>[rank-one quadratic forms]
$$
P_1=ac,\qquad
P_2=bd,\qquad
P_3=(a+b)(c+d).
$$
The first two already give $ac$ and $bd$, while
$$
ad+bc=P_3-P_1-P_2.
$$
Hence all three required polynomials lie in their linear span.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 124 / 3 / i / Solution

A rank-one quadratic form is the square of a <linear form> up to a scalar. More generally, a bilinear expression $(u^Tx)(v^Ty)$ is rank one because its coefficient matrix is the <outer product> $uv^T$.

= Rank
{synonym}

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