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.
The same principles apply for subtraction. For example, consider the difference between the same polynomials given in the example above: $$ \begin{equation}\begin{aligned} &(x^2-3x+4) - (5x^2+3x+7)\\ =&(x^2-5x^2)+(-3x-3x)+(4-7)\\ =&-4x^2-6x-3\\ \end{aligned}\end{equation} $$
Challenge
Find the sums and the differences of the following polynomials:
- $x^3+9x-1$ and $-4x^3-19x+5$
- $2x^2-5$ and $x^3+4x$
- $1-x^4$ and $2x^4-x$
Multiplication
Given two polynomials, we find their product by multiplying each term in the first polynomial by each term in the second polynomial. For example, the product of $x+1$ and $x^2-5$ will involve multiplying each of the $2$ terms in $x+1$ by each of the $2$ terms in $x^2-5$, resulting in $4$ multiplications in total:
$$ \begin{equation}\begin{aligned} &(x+1)(x^2-5)\\ &=x\times x^2+x\times -5+1\times x^2+1\times-5\\ &=x^3-5x+x^2-5\\ &=x^3+x^2-5x-5\\ \end{aligned}\end{equation} $$Created using natural intelligence