Gọi thương của phép chia đa thức f(x)f(x)cho x−1x−1và cho x+2x+2, theo thứ tự là A(x),B(x)A(x),B(x)và dư theo thứ tự là  44 và  11

Ta có:

(x−1).A(x)+4"" class="MathJax_CHTML mjx-chtml" style="box-sizing: inherit; font-style: normal; font-weight: normal; display: inline-block; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-size: 16px; letter-spacing: normal; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; margin: 0px; padding: 1px 0px;">f(x)=(x−1).A(x)+4f(x)=(x−1).A(x)+4

nên (x−1)(x+2).A(x)+4(x+2)"" class="MathJax_CHTML mjx-chtml" style="box-sizing: inherit; font-style: normal; font-weight: normal; display: inline-block; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-size: 16px; letter-spacing: normal; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; margin: 0px; padding: 1px 0px;">(x+2)f(x)=(x−1)(x+2).A(x)+4(x+2)(x+2)f(x)=(x−1)(x+2).A(x)+4(x+2) (1)(1)

(x+2).B(x)+1"" class="MathJax_CHTML mjx-chtml" style="box-sizing: inherit; font-style: normal; font-weight: normal; display: inline-block; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-size: 16px; letter-spacing: normal; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; margin: 0px; padding: 1px 0px;">f(x)=(x+2).B(x)+1f(x)=(x+2).B(x)+1 

nên (x+2)(x−1).B(x)+1(x−1)"" class="MathJax_CHTML mjx-chtml" style="box-sizing: inherit; font-style: normal; font-weight: normal; display: inline-block; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-size: 16px; letter-spacing: normal; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; margin: 0px; padding: 1px 0px;">(x−1)f(x)=(x+2)(x−1).B(x)+1(x−1)(x−1)f(x)=(x+2)(x−1).B(x)+1(x−1) (2)(2)

Lấy (1)(1)trừ  vế theo vế, ta có:

Do đó: 

5x2(x−1)(x+2)+(x+3)"" class="MathJax_CHTML mjx-chtml" style="box-sizing: inherit; font-style: normal; font-weight: normal; display: inline-block; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-size: 16px; letter-spacing: normal; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; margin: 0px; padding: 1px 0px;">⇔f(x)=5x2(x−1)(x+2)+(x+3)