# Arithmetic with polynomials

There is an analogy between polynomials
and the way we write integers in base 10.
For example, the integer 253
is like the polynomial 2*x*^{2} + 5*x* + 3
(especially if you evaluate it when *x* = 10).
When you add integers,
you can arrange the digits in columns and add the columns;
when you add polynomials,
you can arrange the terms in columns and add the columns.
When you multiply integers,
you need to multiply each digit in one integer by each digit in the other;
when you multiply polynomials,
you need to multiply each term in one polynomial by each term in the other.
(In some ways, arithmetic with polynomials is easier to understand,
since there is no carrying.)
For example,
adding *x* + 1 and 2*x* + 3 to get 3*x* + 4
is like adding 11 and 23 to get 34:

1 | 1 | | *x* | + 1 |

2 | 3 | | 2*x* | + 3 |

3 | 4 | | 3*x* | + 4 |

Similarly,
multiplying *x* + 1 and 2*x* + 3
to get 2*x*^{2} + 5*x* + 3
is like multiplying 11 and 23 to get 253:
| 1 | 1 | | | *x* | + 1 |

| 2 | 3 | | | 2*x* | + 3 |

| 3 | 3 | | | 3*x* | + 3 |

2 | 2 | 0 | | 2*x*^{2} | + 2*x* | |

2 | 5 | 3 | | 2*x*^{2} | + 2*x* | + 3 |

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last edited on 2017 August 22.
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