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  "body": "\n## Statement\n\nLet $R$ be an [[integral domain|09S3PFBEPM]] and let $f, g \\in R[x]$.\nThen $\\deg(f g) = \\deg f + \\deg g$, with the [[convention|0DPF6WRZW6]] $\\deg 0 = -\\infty$.\n\n## Remarks\n\nThe proof: if $f$ and $g$ are nonzero with [[leading coefficients|0P05RJC5XA]] $a$ and $b$, then the coefficient of $x^{\\deg f + \\deg g}$ in $f g$ is $a b \\neq 0$.\nOver every ring, $\\deg(f g) \\le \\deg f + \\deg g$.\nThe equality fails in $(\\mathbb{Z}/4\\mathbb{Z})[x]$: $(2x + 1)(2x + 1) = 4x^2 + 4x + 1 = 1$.\n\n## Explanations\n\n### L1 human\n\nWhen you multiply two polynomials, the highest terms multiply to give the highest term of the product.\nFor example, $(x^2 + 1)(x^3 - x)$ has degree $2 + 3 = 5$.\nThis needs the product of the leading coefficients to be nonzero.\n",
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