Operations on Polynomial Functions

Adding and subtracting polynomials

It turns out that polynomials are like numbers, and we can do arithmetic with them!

Let's start with addition of polynomials. If you have two copies of the same term, you can just add them together. Let's do an example to demonstrate this!

Suppose $f(x) = x^2 + 2x + 1$ and $g(x) = 2x^2 + 3x + 4$. Then we see that we can add the $x^2$ terms together, the $x$ terms together, and the constant terms together, just like numbers! Then $$(f+g)(x) = (1 + 2)x^2 + (2+3)x + (1 + 4) = 3x^2 + 5x + 5.$$

Polynomials are the functions we can write in terms of addition and multiplication (of real numbers and variables). For now, we'll stick to polynomials with only one variable.

Because multiplication distributes over addition, we can write polynomials in many ways. All polynomials can be written in the standard form: $$P(x) = a_n x^n + \dots + a_2 x^2 + a_1 x + a_0.$$ Here, $a_0, a_1, \dots, a_n$ are constants (fixed numbers not depending on $x$). We call the highest power, $n$, the degree of the polynomial. The expressions $a_0$, $a_1 x$, $a_2 x^2$, and so on are the terms of the polynomial.

Remember: always keep track of your coefficients and variables carefully!

It's often convenient to keep polynomials reduced to standard form as we work with them. So we'll talk about how to add and multiply polynomials in standard form, and then reduce them back to standard form by combining like terms.

Adding polynomials

Adding polynomials is exactly like adding numbers with several digits, only easier. For example, to add $2x^2 +3x + 1$ and $5x^2-7x+4$: [ \begin{array}{rrrr} & 2x^2 & +3x & +1 \

• & 5x^2 & -7x & +4 \ \hline & 7x^2 & -4x & +5 \end{array} ] Things are even easier because never don't need to "carry a one". We just add the terms with $x^2$, and also the terms with $x$, and also the terms with no variable at all.

You have to be slightly careful when lining things up. Try adding together $t^2 + 4$ and $t - 1$.

You should get: [ \begin{array}{rrrr} & t^2 & & +4 \

• & & t & -1 \ \hline & t^2 & +t & +3 \end{array} ] The $t$ and the $t^2$ have different degrees, so they belong in separate columns.

With practice you can stop writing out the long addition table, just as with adding numbers.

You'll notice that you "distributed" the negative sign using the distributive property!

Multiplying polynomials

Multiplying, as usual, is slightly trickier than adding.

Multiplying a polynomial by a monomial

We'll begin with the special case where one polynomial is a monomial (it only has one term). Try simplifying this example: $$(x^3 + 1) \cdot 2x^2$$

The rule is that multiplication distributes over addition, and that exponents add: $x^a \cdot x^b = x^{a+b}$. So we get $$(x^3+1) \cdot 2x^2 = (x^3 \cdot 2x^2) + (1 \cdot 2x^2) = 2x^5 + 2x^2.$$

Multiplying two polynomials

This is similar to long multiplication. (I bet you can't wait until we get to long division!) For example, to multiply $(x+3) \cdot (2x+1)$: [ \begin{array}{rrrr} & & x & +3 \ \times & & 2x & +1 \ \hline & & x & +3 \

•   & 2x^2 & +6x &    \\
  

\hline & 2x^2 & +7x & +3 \end{array} ] Essentially, we expand out $(x+3)\cdot(2x+1)$ as $(x+3)\cdot 2x + (x+3)\cdot 1$. Then we just multiply monomials: $(x+3)\cdot 1 = x+3$, and $(x+3)\cdot 2x = 2x^2 + 6x$. Finally, we align the terms and add them up.

Also, notice the similarity between this product and $13 \cdot 21 = 273$. (In general, products of polynomials and of numbers don't look the same, because with numbers, you have to carry. But when you don't carry, because the digits are all small, you get the same answer.)

As with addition, eventually you pick up the trick of doing the math without writing out the long multiplication in full form.

This method works for any type of polynomial. For binomials (polynomials with two terms), there's a quicker method called FOIL. To take the product of $(a+b)(c+d)$, where $a, b, c, d$ can be any terms, you add $ac$ (the product of the first terms), $ad$ (the product of the outer terms), $bc$ (the product of the inner terms), and $bd$ (the product of the last terms).

Let's try this with the same problem: To find $(x+3) \cdot (2x+1)$, we take $$\begin{align*} (x+3)(2x+1) &= (x \cdot 2x) + (x \cdot 1) + (3 \cdot 2x) + (3 \cdot 1) \\ &= 2x^2 + x + 6x + 3 \\ &= 2x^2 + 7x + 3. \end{align*}$$

Dividing Polynomials

Polynomial division is a little more complicated, so we will tackle that in another lesson! Division requires a lot more long division.

(Video) Adding, Subtracting, Multiplying, and Simplifying Polynomials - Part 2

by PatrickJMT

Patrick JMT has a good video explaining how to subtract polynomials (complete with a few examples).

Summary

Let's walk through the first example together!

$$\begin{align} [8a^4 + (a-3) - a^2]-[4a^4+2(a+1)+a^2] \\ [8a^4 + a-3 - a^2]-[4a^4+2a+2+a^2] \\ 8a^4 + a-3 - a^2-4a^4-2a-2+a^2 \\ 4a^4 - 2a^2-1a-5 \end{align}$$

We used a combination of distribution and combining like terms to simplify these polynomials. Try the next examples with the video!

Function Operations

Just like we can add, subtract, and multiply polynomials, we can do the same with polynomial functions. A function is a rule that takes an input, does something to it, and gives back an output. If we have two functions, $f(x)$ and $g(x)$, we can perform operations on them just like we do with regular numbers or polynomials.

There are four main operations.

####Addition

To add two functions, we add their corresponding outputs.

$$(f+g)(x) = f(x) + g(x)$$

For example, if $f(x) = 2x+3$ and $g(x) = x^2 -4$, then $(f+g)(x) = f(x) + g(x) = (2x+3) + (x^2 -4) = x^2+2x-1$.

####Subtraction

To subtract functions, subtract their outputs.

$$(f-g)(x) = f(x) - g(x)$$

Using the same functions from the previous example, $(f-g)(x) = f(x) - g(x) = (2x+3) - (x^2 -4) = -x^2+2x+7$.

#####Multiplication

When multiplying functions, you multiply their outputs.

$$(fg)(x) = f(x) \cdot g(x)$$

If we continue using the previous example, we get $(f \cdot g)(x) = f(x) \cdot g(x) = (2x+3)(x^-4) = 2x^3 - 8x + 3x^2 -12$.

#####Division

We can also divide functions, but the result will be a rational function (a fraction of two polynomials).

$$(\frac{f}{g})(x) = \frac{f(x)}{g(x)} = \frac{2x+3}{x^2-4}$$

However, this function might not be defined for certain values of $x$ (like when $g(x)=0$).

In the next lesson, you will learn how to find the degree of a polynomial.