In this section we * learn how to multiply one polynomial function by another polynomial*. The

For instance, by the end of this section we'll know how to quickly write all of the terms in the product:
\[\begin{pmatrix} 3x^5 - 2x^3 + x^2 - 3x + 10\end{pmatrix} . \begin{pmatrix} 2x^4 + 3x^2 - 7x - 5 \end{pmatrix} \]
The *method is explained* in the following ** tutorial**.

We learn the ** method for multiplying two polynomials together** by working through an

- Using a two-way table and simplifying as much as possible, write all the terms in the expansion of the product: \[\begin{pmatrix}2x^4 - x^3 + 3x^2+4x - 5 \end{pmatrix}.\begin{pmatrix}x^3 - 2x^2 + 3x - 4\end{pmatrix}\]
- Given \(f(x) = 3x^5 - 2x^3+x^2 - x + 4\) and \(g(x) = 2x^4 + x^3 - 3x + 1\), find the product: \[f(x)\times g(x)\]
- Using a two-way table and simplifying as much as possible, write all the terms in the expansion of the product: \[\begin{pmatrix} -2x^6 + 3x^4 - 5x^3 + x^2 - x + 3 \end{pmatrix}.\begin{pmatrix} 2x^3+ 5x^2 - x +4\end{pmatrix}\]
- Given \(p(x) = \frac{x^4}{2} - 3x^3 + 2x^2 + 5x - 1\) and \(q(x) = 6x^4 + 4x^2 - 8x + 2\), find the product: \[p(x)\times q(x)\].
- Using a two-way table and simplifying as much as possible, write all the terms in the expansion of the product: \[\begin{pmatrix} 6x^4 - 3x^2 + x - 5 \end{pmatrix}.\begin{pmatrix} 2x^3+ 4x^2 - 7x - 2 \end{pmatrix}\]

- For all the terms in the expansion of the product: \[\begin{pmatrix}2x^4 - x^3 + 3x^2+4x - 5 \end{pmatrix}.\begin{pmatrix}x^3 - 2x^2 + 3x - 4\end{pmatrix}\] we find: \[2x^7 - 5x^6 + 11x^5 - 13x^4 + 10x^2 - 31x + 20\]
- Given \(f(x) = 3x^5 - 2x^3+x^2 - x + 4\) and \(g(x) = 2x^4 + x^3 - 3x + 1\), we find: \[f(x)\times g(x) = 6x^9 + 38x^8 - 4x^7 - 9x^6 + 2x^5 + 13x^4 - x^3 + 4x^2 - 13x + 4\]
- For all the terms in the expansion of the product: \[\begin{pmatrix} -2x^6 + 3x^4 - 5x^3 + x^2 - x + 3 \end{pmatrix}.\begin{pmatrix} 2x^3+ 5x^2 - x +4\end{pmatrix}\] we find: \[-4x^9 - 10x^8 + 8x^7 - 3x^6 - 26x^5 + 20x^4 - 20x^3 + 20x^2 - 7x + 12\]
- Given \(p(x) = \frac{x^4}{2} - 3x^3 + 2x^2 + 5x - 1\) and \(q(x) = 6x^4 + 4x^2 - 8x + 2\), we find: \[p(x)\times q(x) = 3x^8 - 18x^7 + 14x^6 + 14x^5 + 27x^4 - 2x^3 - 40x^2 + 18x - 2\].
- For all the terms in the expansion of the product: \[\begin{pmatrix} 6x^4 - 3x^2 + x - 5 \end{pmatrix}.\begin{pmatrix} 2x^3+ 4x^2 - 7x - 2 \end{pmatrix}\] we find: \[12x^7 + 24x^6 - 48x^5 - 24x^4 + 15x^3 - 21x^2 + 33x + 10\]

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