Polynomials

A polynomial is an expression containing the variable x and the standard operations of addition, multiplication and raising to a non-negative integer power. We can add, subtract, multiply, divide and raise polynomials to a power.

Basic ideas

A simple example of a polynomial is

$2 x^{2} + 5 x - 12$

In general we can write a polynomial like this:

$a_{n} x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{1} x^{1} + a_{0} x^{0},$

where the real numbers in front of x, that is a_n, are called coefficients, and n is called the degree of the polynomial. The number n is the highest power in the polynomial for which a_n ≠ 0. If a_n were zero, we would in effect get rid of the variable x that the coefficient belongs to, because zero times anything is zero:

$0 \cdot x^{3} = 0.$

So the first polynomial above has degree two, because the highest power of x with a non-zero coefficient is two. For another polynomial,

$7 x^{2} + 5 x^{4} + 15 x$

the degree is 4 (the highest index is 4). We usually write polynomials in descending order of powers, so that the term with the highest power comes first. We can therefore rewrite the previous polynomial like this:

$5 x^{4} + 7 x^{2} + 15 x .$

Notice that the polynomial has no terms where the variable has index three or zero. That does not matter at all: we simply imagine that the coefficients of these terms are zero. We could write

$5 x^{4} + 0 x^{3} + 7 x^{2} + 15 x^{1} + 0 x^{0} .$

Now every term is there, but the polynomial is needlessly cluttered, so we usually leave out the terms with zero coefficients.

Adding and subtracting polynomials

Adding and subtracting polynomials is fairly straightforward. We just add or subtract the coefficients of terms with the same power. So

$a x^{n} + b x^{n} = (a + b) x^{n} .$

A simple example:

$3 x^{2} + 5 x^{2} = (3 + 5) x^{2} = 8 x^{2}$

To add longer polynomials, we always have to collect the terms with the same power of the variable.

$\begin{array}{r} (7 x^{3} + 5 x^{2} + x) + (2 x^{3} - 3 x^{2} + 9 x) = \\ = (7 + 2) x^{3} + (5 - 3) x^{2} + (1 + 9) x . \end{array}$

If a term with a given power does not appear in one of the polynomials, it stays unchanged (imagine that the other polynomial has such a term, but with a zero coefficient).

$(x^{3} + 2 x) + (4 x^{2} + 5 x + 3) = x^{3} + 4 x^{2} + 7 x + 3$

Subtracting polynomials works just like adding them, except that we first multiply the second polynomial by minus one, that is, we change all the signs. Here is an example:

$(3 x^{2} + 6 x) - (2 x^{2} + 14 x) = (3 x^{2} + 6 x) + (- 2 x^{2} - 14 x)$

That is now ordinary addition of polynomials, so we can add as usual:

$(3 x^{2} + 6 x) + (- 2 x^{2} - 14 x) = x^{2} - 8 x$

Remember that a negative term in the second polynomial becomes positive after this step.

$\begin{array}{rcl} (x^{2} + 2 x) - (3 x^{2} - 10 x) = \\ = & (x^{2} + 2 x) + (- 3 x^{2} + 10 x) = \\ = & - 2 x^{2} + 12 x \end{array}$

Multiplying polynomials

To multiply polynomials, we multiply every term of the first polynomial by every term of the second. We multiply the coefficients as ordinary real numbers, but we simply add the indices of the variables, following the rules for powers. Here is an example:

$(3 x^{2} + 4 x) \cdot 5 x^{2} = (3 \cdot 5) x^{2 + 2} + (4 \cdot 5) x^{1 + 2} = 15 x^{4} + 20 x^{3}$

This example multiplied by a single term, so it did not show clearly how every term is multiplied by every other. Here is a slightly bigger example:

$\begin{array}{rcl} (x^{3} + 2 x) \cdot (4 x^{2} + 7 x) = \\ = & (1 \cdot 4) x^{3 + 2} + (1 \cdot 7) x^{3 + 1} + (2 \cdot 4) x^{1 + 2} + (2 \cdot 7) x^{1 + 1} \end{array}$

If there is a minus sign in a polynomial, it carries through into the multiplication as usual.

$\begin{array}{rcl} (x^{3} - 2 x) \cdot (4 x^{2} - 7 x) = \\ = & (1 \cdot 4) x^{3 + 2} + (1 \cdot (- 7)) x^{3 + 1} + (- 2 \cdot 4) x^{1 + 2} + \\ + & (- 2 \cdot (- 7)) x^{1 + 1} = 4 x^{5} - 7 x^{4} - 8 x^{3} + 14 x^{2} \end{array}$

Simplifying polynomials

When we simplify polynomials, we usually want to make the expression simpler. For that we use things like writing equivalent fractions, cancelling, taking out common factors, applying formulae and so on. It is a good idea to know these formulae. Here we will use some of them, and I will mention them as we go. A classic formula is

$(a + b)^{2} = a^{2} + 2 a b + b^{2}$

If you wonder where this formula comes from, just expand the brackets.

$\begin{array}{rcl} (a + b)^{2} & = & (a + b) (a + b) \\ = & a^{2} + a b + b a + b^{2} \\ = & a^{2} + 2 a b + b^{2} \end{array}$

Let’s start with a simple example.

$\begin{array}{rcl} (a + b)^{2} + (a + 2 b)^{2} - 3 a \cdot 2 b = \\ = & a^{2} + 2 a b + b^{2} + a^{2} + 4 a b + 4 b^{2} - 6 a b = \\ = & 2 a^{2} + 5 b^{2} \end{array}$

We expanded both squared brackets using the formula above. Remember that we expanded the second bracket as

$(a + 2 b)^{2} = a^{2} + 4 a b + 4 b^{2} .$

Another classic formula is

$a^{2} - b^{2} = (a + b) (a - b) .$

The second example is an algebraic fraction. The numerator and denominator are both polynomials, and we will try to rewrite them so that the whole fraction cancels nicely. In the first step we apply the first formula to the numerator, but in reverse. In the denominator we apply the second formula. At the end we simply cancel (a + b).

$\begin{array}{rcl} \frac{a^{2} + 2 a b + b^{2}}{a^{2} - b^{2}} & = & \frac{(a + b)^{2}}{(a + b) (a - b)} \\ = & \frac{(a + b) (a + b)}{(a + b) (a - b)} \\ = & \frac{a + b}{a - b} \end{array}$

In the third example we try taking out a common factor. In the numerator we first take out 3a³ and then factorise the bracket (4a² − 1) using the difference of two squares a² − b², making use of the fact that

$1^{2} = 1^{1} \to (a^{2} - 1^{2}) = (a + 1) (a - 1) .$

In the denominator we first simply multiply the first two terms, then take out 3a³ too, and finally cancel everything.

$\begin{array}{rcl} \frac{12 a^{5} - 3 a^{3}}{2 a^{2} \cdot 3 a^{2} + 3 a^{3}} & = & \frac{3 a^{3} (4 a^{2} - 1)}{6 a^{4} + 3 a^{3}} \\ = & \frac{3 a^{3} (2 a + 1) (2 a - 1)}{3 a^{3} (2 a + 1)} \\ = & 2 a - 1 \end{array}$

Chapters: Polynomials, Dividing a polynomial by a polynomial, Root of a polynomial, Factorising polynomials

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