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In a covalent molecule, can we conclude that if the formal charge between two atoms are +1 and -1 respectively, then the chemical bond acting between them is a polar covalent bond (such as the bond between N and single-bonded O in Nitrate Ion)? On the other hand, if both of the atoms has a formal charge of 0, then the chemical bond acting between them is a nonpolar covalent bond (bonds in sulfur dioxide and sulfur trioxide)? How about bond between atoms with formal charge of +1 and 0 (P with single-bonded O in phosphate ion)?

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All bonds between two atoms of different elements are polar to some extent. To determine how polar, you need to check the elements’ electronegativities and only them.

Formal charges are exactly what their name implies: a formality. When drawing Lewis structures you should attempt to generate all-octet structures but often this isn’t possible without moving electrons entirely between atoms. In that case, formal charges are assigned to rationalise e.g. a nitrogen with only four electrons.

Image of nitric acid, sulphur trioxide and carbon monoxide

Check out these structures of nitric acid, sulphur trioxide and carbon monoxide. (Resonance structures applicable to $\ce{HNO3}$ and $\ce{SO3}$ not displayed.) All bonds are polarised towards the oxygen, since oxygen’s electronegativity is $3.5$; the second highest amoung the non-noble-gas elements. Note especially that the bonds in $\ce{CO}$ are polarised towards oxygen despite oxygen bearing a formal positive charge. (There is a lot more to $\ce{CO}$ which is shown well in Martin’s answer including the molecular orbitals)

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