Example 1
Divide \( x^3 + 3x^3 + 2x + 1 \) by \( x+2 \).
\( \require{AMSsymbols} \require{enclose} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \bbox[yellow]{x^2} \\ \bbox[orange]{x}+2 \enclose{longdiv}{\bbox[pink]{x^3} +4x^2 +2x +1} \cdots \bbox[orange]{x} \times \bbox[yellow]{x^2} = \bbox[pink]{x^3} \)
\( \require{AMSsymbols} \require{enclose} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \bbox[yellow]{x^2} \\ \bbox[orange]{x+2} \enclose{longdiv}{x^3 +4x^2 +2x +1} \\ \ \ \ \ \ \ \ \ \ \ \ \underline{\bbox[pink]{x^3+2x^2}} \cdots \bbox[orange]{(x+2)} \times \bbox[yellow]{x^2} = \bbox[pink]{x^3+2x^2} \)
\( \require{AMSsymbols} \require{enclose} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x^2 \\ x+2 \enclose{longdiv}{x^3 + \bbox[yellow]{4x^2} +2x +1} \\ \ \ \ \ \ \ \ \ \ \ \ \underline{x^3+ \bbox[yellow]{2x^2}} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \bbox[yellow]{2x^2}+2x+1 \cdots \bbox[yellow]{4x^2-2x^2 = 2x^2} \)
\( \require{AMSsymbols} \require{enclose} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x^2 + \bbox[yellow]{2x} \\ \bbox[orange]{x}+2 \enclose{longdiv}{x^3 +4x^2 +2x +1} \\ \ \ \ \ \ \ \ \ \ \ \ \underline{x^3+2x^2} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \bbox[pink]{2x^2} + 2x+1 \cdots \bbox[orange]{x} \times \bbox[yellow]2x = \bbox[pink]{2x^2} \)
\( \require{AMSsymbols} \require{enclose} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x^2 + \bbox[yellow]{2x} \\ \bbox[orange]{x+2} \enclose{longdiv}{x^3 +4x^2 +2x +1} \\ \ \ \ \ \ \ \ \ \ \ \ \underline{x^3+2x^2} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2x^2+2x+1 \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \underline{\bbox[pink]{2x^2+4x}} \cdots \bbox[orange]{(x+2)} \times \bbox[yellow]{2x} = \bbox[pink]{2x^2 + 4x} \)
\( \require{AMSsymbols} \require{enclose} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x^2 +2x \\ x+2 \enclose{longdiv}{x^3 +4x^2 +2x +1} \\ \ \ \ \ \ \ \ \ \ \ \ \underline{x^3+2x^2} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \bbox[yellow]{2x^2+2x+1} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \underline{\bbox[yellow]{2x^2+4x}} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \bbox[yellow]{-2x+1} \cdots \bbox[yellow]{(2x^2+2x+1)-(2x^2+4x) = -2x+1} \)
\( \require{AMSsymbols} \require{enclose} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x^2 +2x \bbox[yellow]{-2} \\ \bbox[orange]{x}+2 \enclose{longdiv}{x^3 +4x^2 +2x +1} \\ \ \ \ \ \ \ \ \ \ \ \ \underline{x^3+2x^2} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2x^2+2x+1 \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \underline{2x^2+4x} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \bbox[pink]{-2x}+1 \cdots \bbox[yellow]{-2} \times \bbox[orange]{x} = \bbox[pink]{-2x} \)
\( \require{AMSsymbols} \require{enclose} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x^2 +2x \bbox[yellow]{-2} \\ \bbox[orange]{x+2} \enclose{longdiv}{x^3 +4x^2 +2x +1} \\ \ \ \ \ \ \ \ \ \ \ \ \underline{x^3+2x^2} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2x^2+2x+1 \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \underline{2x^2+4x} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -2x+1 \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \underline{\bbox[pink]{-2x-4}} \cdots \bbox[yellow]{-}2 \times \bbox[orange]{(x+2)} = \bbox[pink]{-2x-4} \)
\( \require{AMSsymbols} \require{enclose} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x^2+2x-2 \\ x+2 \enclose{longdiv}{x^3 +4x^2 +2x +1} \\ \ \ \ \ \ \ \ \ \ \ \ \underline{x^3+2x^2} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2x^2+2x+1 \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \underline{2x^2+4x} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \bbox[yellow]{-2x+1} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \underline{\bbox[yellow]{-2x-4}} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \bbox[yellow]{5} \cdots \bbox[yellow]{(-2x+1)-(-2x-4) = 5} \)
Example 2
Perform a long division: \( (4x^4-6x^3+2x^2-3x+5) \div (2x+1) \).
\( \require{AMSsymbols} \require{enclose} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \bbox[yellow]{2x^3} \\ \bbox[orange]{2x}+1 \enclose{longdiv} {\bbox[pink]{4x^4}-6x^3 + 2x^2-3x + 5} \cdots \bbox[pink]{4x^4} \div \bbox[orange]{2x} = \bbox[yellow]{2x^3} \)
\( \require{AMSsymbols} \require{enclose} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \bbox[yellow]{2x^3} \\ \bbox[orange]{2x+1} \enclose{longdiv} {4x^4-6x^3 + 2x^2-3x + 5} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \underline{\bbox[pink]{4x^4+2x^3}} \leftarrow \bbox[orange]{(2x+1)} \times \bbox[yellow]{2x^3} \)
\( \require{AMSsymbols} \require{enclose} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2x^3 \\ 2x+1 \enclose{longdiv} {4x^4 \bbox[yellow]{-6x^3} + 2x^2-3x + 5} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \underline{4x^4 \bbox[orange]{+2x^3}} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \bbox[pink]{-8x^3} + 2x^2 \cdots \bbox[yellow]{(-6x^3)}-\bbox[orange]{2x^3} = \bbox[pink]{-8x^3} \)
\( \require{AMSsymbols} \require{enclose} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2x^3 \bbox[yellow]{-4x^2} \\ \bbox[orange]{2x}+1 \enclose{longdiv} {4x^4-6x^3 + 2x^2-3x + 5} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \underline{4x^4+2x^3} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \bbox[pink]{-8x^3} + 2x^2 \cdots \bbox[pink]{-8x^3} \div \bbox[orange]{2x} = \bbox[yellow]{- 4x^2} \)
\( \require{AMSsymbols} \require{enclose} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2x^3 \bbox[yellow]{-4x^2} \\ \bbox[orange]{2x+1} \enclose{longdiv} {4x^4-6x^3 + 2x^2-3x + 5} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \underline{4x^4+2x^3} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -8x^3+2x^2 \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \underline{\bbox[pink]{-8x^3-4x^2}} \leftarrow \bbox[orange]{(2x+1)} \times \bbox[yellow]{-4x^2} = \bbox[pink]{-8x^3-4x^2} \)
\( \require{AMSsymbols} \require{enclose} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2x^3-4x^2 \\ 2x+1 \enclose{longdiv} {4x^4-6x^3 + 2x^2-3x + 5} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \underline{4x^4+2x^3} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -8x^3 \bbox[yellow]{+2x^2} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \underline{-8x^3 \bbox[orange]{-4x^2}} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \bbox[pink]{6x^2}-3x \cdots \bbox[yellow]{+2x^2}-\bbox[orange]{-4x^2} = \bbox[pink]{6x^2} \)
\( \require{AMSsymbols} \require{enclose} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2x^3-4x^2 + \bbox[yellow]{3x} \\ \bbox[orange]{2x}+1 \enclose{longdiv} {4x^4-6x^3 + 2x^2-3x + 5} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \underline{4x^4+2x^3} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -8x^3+2x^2 \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \underline{-8x^3-4x^2} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \bbox[pink]{6x^2}-3x \cdots \bbox[pink]{6x^2} \div \bbox[orange]{2x} = \bbox[yellow]{3x} \)
\( \require{AMSsymbols} \require{enclose} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2x^3-4x^2 + \bbox[yellow]{3x} \\ \bbox[orange]{2x+1} \enclose{longdiv} {4x^4-6x^3 + 2x^2-3x + 5} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \underline{4x^4+2x^3} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -8x^3+2x^2 \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \underline{-8x^3-4x^2} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 6x^2-3x \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \underline{\bbox[pink]{6x^2+3x}} \leftarrow \bbox[orange]{(2x+1)} \times \bbox[yellow]{3x} \)
\( \require{AMSsymbols} \require{enclose} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2x^3-4x^2 + 3x \\ 2x+1 \enclose{longdiv} {4x^4-6x^3 + 2x^2-3x + 5} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \underline{4x^4+2x^3} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -8x^3+2x^2 \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \underline{-8x^3-4x^2} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 6x^2 \bbox[yellow]{-3x} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \underline{6x^2 \bbox[orange]{+3x}} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \bbox[pink]{-6x}+5 \cdots \bbox[yellow]{-3x}-\bbox[orange]{+3x} = \bbox[pink]{-6x} \)
\( \require{AMSsymbols} \require{enclose} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2x^3-4x^2 + 3x \bbox[yellow]{-3} \\ \bbox[orange]{2x} + 1 \enclose{longdiv} {4x^4-6x^3 + 2x^2-3x + 5} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \underline{4x^4+2x^3} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -8x^3+2x^2 \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \underline{-8x^3-4x^2} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 6x^2-3x \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \underline{6x^2+3x} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \bbox[pink]{-6x}+5 \cdots \bbox[pink]{-6x} \div \bbox[orange]{2x} = \bbox[yellow]{-3} \)
\( \require{AMSsymbols} \require{enclose} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2x^3-4x^2 + 3x \bbox[yellow]{-3} \\ \bbox[orange]{2x+1} \enclose{longdiv} {4x^4-6x^3 + 2x^2-3x + 5} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \underline{4x^4+2x^3} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -8x^3+2x^2 \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \underline{-8x^3-4x^2} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 6x^2-3x \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \underline{6x^2+3x} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -6x+5 \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \underline{\bbox[pink]{-6x-3}} \leftarrow \bbox[orange]{(2x+1)} \times \bbox[yellow]{-3} \)
\( \require{AMSsymbols} \require{enclose} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2x^3-4x^2 + 3x-3 \\ 2x+1 \enclose{longdiv} {4x^4-6x^3 + 2x^2-3x + 5} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \underline{4x^4+2x^3} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -8x^3+2x^2 \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \underline{-8x^3-4x^2} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 6x^2-3x \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \underline{6x^2+3x} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -6x \bbox[yellow]{+5} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \underline{-6x \bbox[orange]{-3}} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \bbox[pink]{8} \leftarrow \bbox[yellow]{+5} – \bbox[orange]{-3} \)
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