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# Polynomial

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Simplify the following:
(2y-1)(4y^{10}+y^9+y^8+y^7).
Express your answer as a polynomial with the degrees of the terms in decreasing order.

May 9, 2022

#1
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Just expand it using tabular method:

You list the terms of the first polynomial in the columns, and the terms of the second polynomial in the rows:

$$\begin{array}{|c|c|c|} \hline &2y&-1\\ \hline 4y^{10}&&\\\hline +y^9&&\\\hline +y^8&&\\\hline +y^7&&\\\hline \end{array}$$

And then fill in the table with the product of row and column. For example, 4y^{10} * 2y = 8y^{11}, so:

$$\begin{array}{|c|c|c|} \hline &2y&-1\\ \hline 4y^{10}&\color{blue}8y^{11}&\\\hline +y^9&&\\\hline +y^8&&\\\hline +y^7&&\\\hline \end{array}$$

Filling in the table like so:

$$\begin{array}{|c|c|c|} \hline &2y&-1\\ \hline 4y^{10}&\color{blue}8y^{11}&\color{blue}-4y^{10}\\\hline +y^9&\color{blue}2y^{10}&\color{blue}-y^9\\\hline +y^8&\color{blue}2y^9&\color{blue}-y^8\\\hline +y^7&\color{blue}2y^8&\color{blue}-y^7\\\hline \end{array}$$

To get the product, add all the blue terms in the table:

$$(2y - 1)(4y^{10} + y^9 + y^8 + y^7) = 8y^{11} - 4y^{10} + 2y^{10} - y^9 + 2y^9 - y^8 + 2y^8 - y^7$$