Polynomial Arithmetic

We can add, subtract, multiply polynomials. Polynomials can involve many variables raised to whole number powers so for simplicity we will be using polynomials only involving the variable $x$.

Addition and subtraction

When adding/subtracting polynomials, we simply add/subtract the terms with the same power of $x$.

For example, let’s add the polynomials $x^2-3x+4$ and $5x^2+3x+7$. To make visualizing the addition easier, we will be using brackets: $$ \begin{equation}\begin{aligned} &(x^2-3x+4) + (5x^2+3x+7)\\ =&(x^2+5x^2)+(-3x+3x)+(4+7)\\ =&6x^2+0x+11\\ =&6x^2+11\\ \end{aligned}\end{equation} $$

For a regular addition, we do not need to specify the brackets or reorder the terms. These were just done to make emphasis in the demonstration above.

Challenge

Add the following polynomials:

• $x^3+9x-1$ and $-4x^3-19x+5$
• $2x^2-5$ and $x^3+4x$
• $1-x^4$ and $2x^4-x$

The same principles apply for subtraction. For example, consider the difference between the same polynomials given in the example above: $$ \begin{equation}\begin{aligned} &(x^2-3x+4) - (5x^2+3x+7)\\ =&(x^2-5x^2)+(-3x-3x)+(4-7)\\ =&-4x^2-6x-3\\ \end{aligned}\end{equation} $$

Multiplication

When multiplying two polynomials, we need to find the sum of the products of each term in the first with each term in the second. For example, consider two binomials, $x-2$ and $x^2+1$. Each has $2$ terms and thus multiplying the two polynomials would involve $2\times 2$ multiplications - each term in the first, $x$ and $-2$, by each term in the second, $x^2$ and $1$. Again, brackets are used for emphasis:

$$ \begin{equation}\begin{aligned} &(x-2)(x^2+1)\\ =&(x\times x^2)+(x\times 1)+(-2\times x^2)+(-2\times 1)\\ =&x^3+x-2x^2-2\\ =&x^3-2x^2+x-2\\ \end{aligned}\end{equation} $$

Note that the final line was obtained by rearranging the terms according to the commutative and associative properties of multiplication.

Challenge

Multiply the following polynomials:

• $x-1$ and $x+3$ (hint: $4$ products)
• $4x^2+5x+3$ and $x-5$ (hint: $6$ products)
• $5-x^2$ and $x^2+2x+1$ (hint: $6$ products)

Division

Long division of polynomials involves the same steps as long division of integers:

• Divide by the first term in the divisor
• Multiply by the entire divisor
• Subtract
• Bring down the remaining terms

$$ \begin{equation}\begin{aligned} \begin{array}{r} x+2\\ x+3 \overline{) \ \ x^2+5x+7} \\ \underline{-(x^2+3x)}\ \ \ \ \ \\ 2x+7 \\ \underline{-(2x+6)} \\ 1 \end{array} \end{aligned}\end{equation} $$