# Multiplication with Polynomials

## (multiplying polynomial functions together)

In this section we learn how to multiply one polynomial function by another polynomial. The method involves a two-way table and is a more efficient and more reliable (less prone to error) than "simple" distribution.

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.

### Tutorial: multiplying polynomials together

We learn the method for multiplying two polynomials together by working through an example in which we expand and simplify the following: $\begin{pmatrix}2x^5 - 3x^3 + 4x^2 + x - 3 \end{pmatrix}. \begin{pmatrix} x^3 + 2x^2 + 4x - 5 \end{pmatrix}$

## Exercise 1

1. Using a two-way table and simplifying as much as possible, write all the terms in the expansion of the product: $\begin{pmatrix}2x^4 - x^3 + 3x^2+4x - 5 \end{pmatrix}.\begin{pmatrix}x^3 - 2x^2 + 3x - 4\end{pmatrix}$

2. Given $$f(x) = 3x^5 - 2x^3+x^2 - x + 4$$ and $$g(x) = 2x^4 + x^3 - 3x + 1$$, find the product: $f(x)\times g(x)$

3. Using a two-way table and simplifying as much as possible, write all the terms in the expansion of the product: $\begin{pmatrix} -2x^6 + 3x^4 - 5x^3 + x^2 - x + 3 \end{pmatrix}.\begin{pmatrix} 2x^3+ 5x^2 - x +4\end{pmatrix}$

4. 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: $p(x)\times q(x)$.

5. Using a two-way table and simplifying as much as possible, write all the terms in the expansion of the product: $\begin{pmatrix} 6x^4 - 3x^2 + x - 5 \end{pmatrix}.\begin{pmatrix} 2x^3+ 4x^2 - 7x - 2 \end{pmatrix}$

Note: this exercise can be downloaded as a worksheet to practice with:

## Solution Without Working

1. 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: $2x^7 - 5x^6 + 11x^5 - 13x^4 + 10x^2 - 31x + 20$

2. Given $$f(x) = 3x^5 - 2x^3+x^2 - x + 4$$ and $$g(x) = 2x^4 + x^3 - 3x + 1$$, we find: $f(x)\times g(x) = 6x^9 + 38x^8 - 4x^7 - 9x^6 + 2x^5 + 13x^4 - x^3 + 4x^2 - 13x + 4$

3. 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: $-4x^9 - 10x^8 + 8x^7 - 3x^6 - 26x^5 + 20x^4 - 20x^3 + 20x^2 - 7x + 12$

4. Given $$p(x) = \frac{x^4}{2} - 3x^3 + 2x^2 + 5x - 1$$ and $$q(x) = 6x^4 + 4x^2 - 8x + 2$$, we find: $p(x)\times q(x) = 3x^8 - 18x^7 + 14x^6 + 14x^5 + 27x^4 - 2x^3 - 40x^2 + 18x - 2$.

5. 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: $12x^7 + 24x^6 - 48x^5 - 24x^4 + 15x^3 - 21x^2 + 33x + 10$