In this video, we discuss polynomial functions which includes linear and quadratic functions as special cases and power functions which focus on particular powers of the variable x. Square roots and other types of roots arise as fractional powers which might look a bit surprising if you've never seen this before. In fact, we'll start with very simple powers and you'll see the potential to develop ideas in several directions. A power function has the form y equals x to the n, where x is the input, an independent variable which you can choose freely, any some fixed number called an exponent to be used to form the power of x, and y equals x to the n is the output. Let's start off with the simplest cases when the exponent n is a positive integer 1, 2, 3, and so on, and x to the n is x times x times x and so on with n occurrences of x called the nth power of x. For example, when n equals one, we get the so called lazy function y equals x to the one which is just x that simply returns the input as the output without making any changes at all. This is also known as the identity function. When n equals two, we get f of x equals x squared, the square function which you are familiar with already. Where n equals three, we get the cube function, so called because x times x times x is the volume of a cube of side length x. For example, two cubed is eight, minus two cubed is negative eight and negative volume, three cubed is 27 and negative three cubed is negative 27 and so on. If we undo the nth power function, then we get what's called the nth root function whose rule is denoted like this, with a tiny n sitting inside a root symbol, and we say, y is the nth root of x. When n equals two, this becomes the usual square root which you can think of as undoing the square function. When n equals three, it becomes what we call the cube root which you can think of as undoing the cube function. The idea of undoing functions is called inversion in mathematics. An entire video is devoted later to formalizing the concept of an inverse function. Here are some examples of undoing square and cube roots and a couple of fourth roots. For example, two to the fourth is equal to 16. So, if we undo that, we get the fourth root of 16, gets you back to two, and three to the fourth is 81. So, if we undo that, we get the fourth root of 81, gets you back to three. We have a really useful notation for taking roots. The reciprocal of n as an integer is one over n, and for reasons that we discuss later, it's convenient to write the nth root of x as x to the one on n, exploiting the reciprocal n in the exponent. Something rather extraordinary happens. nth root functions turn out to be power functions but using reciprocals of integers as powers. For examples, the square root of two becomes two to the power of one half. The square root of three becomes three to the power of one half, two the square of four becomes four to the half, and also being the cube root of eight is eight to the one third. Three is nine to the half and also 27 to the one third and so on. There are some important laws for manipulating fractional powers and you'll see them in a wider context in a later video when we discuss general exponential laws. For example, the fact that the square root of a product is the product of the square roots converts into an elegant equation involving fractional powers. ab to the half is a to the half multiplied by b to the half. Similarly, we have a over b, all to the half is a to the half divided by b to the half. For now though, I want to return to integer powers of x and combine them in a special way to form what are called polynomials or polynomial functions. These are formed by adding together constant multiples of powers of x. Of course, we have x, x squared, x cubed, and so on, but we also have x to the zero, the zeroth power of x, which is defined to be the constant number one, and then we can talk about combining non-negative powers of x. A polynomial which we typically call p of x, p for polynomial, has the form a_0 plus a_1x plus a_2x squared plus all the way up to a_nx to the n, where n is an integer greater than or equal to zero, called the degree of the polynomial, and a_0 up to a_n are real constants such that a_n is not equal to zero. Note that the polynomial is constructed from x and all the constants using just simple arithmetic; additions and multiplications used even to form powers of x. The simplest cases are, for n equals zero, p of x equals a_0 which is a constant; for n equals one, p of x is a linear function; for n equals two, p of x is a quadratic function; and for n equals three, p of x is called a cubic function because it involves a term with x cubed. In a certain sense, polynomials are the nicest of all functions. You only need basic arithmetic to work with them. For this reason, it's a common thing in mathematics to reduce problems about arbitrary functions to polynomial functions. The key that enables all of this to work turns out to be calculus. It is very important to be able to factorize polynomials related to solving equations. A central fact is the following: Suppose that p of x is a polynomial of degree n, and p of Lambda is equal to zero for some real number Lambda, by which we mean that we end up with zero after evaluating the polynomial expression with x replaced by Lambda. The conclusion is that we can factorize p of x as x minus Lambda times q of x, for some polynomial q of x of degree n minus 1, which is of degree one less than that of p of x. This is substantial progress to be able to reduce the degree of the polynomial by means of the factorization. The reason why this works is explained in the notes. It may look rather abstract but it's really neat because if you know a root of the polynomial, that is a number that causes a polynomial to evaluate to zero, then you can simplify things by factorizing the polynomial. Let's look at some examples. Suppose the p of x is x squared minus 3x plus 2. Observe that p of one evaluates to zero. So, using this fact mentioned earlier, we have the p of x is x minus 1 times q of x, where q of x has degree 2 minus 1 equal to 1. So, q of x is a linear polynomial. Now by inspection, p of x is x minus 1 times x minus 2. So, q of x is, in fact, x minus 2. But sometimes you're unable to see things by inspection, so we have another trick up our sleeves called long division of polynomials. You may not be familiar with this method, so we'll apply it carefully to this previous example. First, you write x squared minus 3x plus 2 inside this long division symbol with x minus 1 outside. The idea is to scoop out multiples of x minus 1 from what's inside the long division symbol trying to reduce things down ideally to make the contents vanish completely. First, we write x on top. On the second line, we write x copies of x minus 1, which is x squared minus x, and then take this away from x squared minus 3x plus 2. Notice when we do this, the x squared terms disappear and what is left over is minus 3x plus 2. Take away minus x which is just minus 2x plus 2. Now, we try to make this in term simplify, so we put minus 2 on top and scoop out minus 2 copies of x minus 1 which is minus 2x plus 2, which we write on the line below. When we take this away, we're left with zero. The whole point is that p of x, our original quadratic inside the long division symbol is just the thing on top, x minus 2 multiplied by x minus 1. So, the thing on top turns out to be our q of x, and p of x equal to x minus 1 q of x just recovers the factorization we noted earlier. Now, let's do a difficult example which is to factorize p of x given by this cubic polynomial. It's not even clear how to get started. Just for exploration, lets start evaluating p of x at some simple inputs for x. For example, we quickly see that p of one evaluates to minus 24, p of two evaluates to minus 21, and p of three evaluates to zero. We're lucky, we've stumbled on a root of p of x namely x equals three. So, we can use the earlier fact that guarantees that x minus 3 is a factor of p of x. Thus, we may write, p of x is x minus 3 times q of x. For some polynomial q of x of degree one less than three which is two, so q of x will be a quadratic. Let's do the long division. First, by writing a long division symbol with p of x inside and x minus 3 outside. Then, we try to make what's inside the long division symbol vanish completely. We begin by scooping out x squared copies of x minus 3, so write x square on top and x squared times x minus 3, which is x cubed minus 3x squared underneath. Taking this away causes the x cubed terms to vanish and we're left with six x squared minus 13x minus 15 at the next line. So, now we try to make this simplify, by writing plus 6x on top and 6x squared minus 18x on the line below. The results of scooping out 6x copies of x minus 3. Now, the x squared terms vanish, when we take this away from the previous line, and we get 5x minus 15. Now, we write plus 5 on top and 5x minus 15 on the line below, scooping out five copies of x minus 3, and when we take this away we're left with zero. That's great because it means the polynomial on top, the quadratic x square plus 6x plus 5 when multiplied by x minus 3 manages to scoop out all of p of x. So, it becomes the q of x we've been looking for. Thus, p of x is x minus 3 times this quadratic q of x. Once we have a quadratic, we feel like we're at home because we have met the factorizing then, either by inspection or if necessary by resorting to the quadratic formula. Here, the quadratic factorizes as x plus 5 times x plus 1, and we get a full factorization of the original cubic into linear factors. If, for example, we needed to solve the equation p of x equals zero, then we would now have a factorization of zero. So, one of the factors has to be zero, either x minus 3, x plus 5, or x plus 1, and it's immediate that x equals 3, minus 5, or minus 1 and we can write down the solution set. Now, before, we stumbled on the root three and then used this method involving long division of polynomials. If we'd stumbled on one of the other roots, minus 5 or minus 1, then we could have used exactly the same method but using a different linear factor in the long division. You can try that out for practice using say, x plus 5 or x plus 1 in the long division and you should end up ultimately with the same factorization of the cubic. We're gradually building up our repertoire of functions, useful tricks, and techniques, all of which will provide us with an excellent foundation for the calculus modules that lie ahead. Today, we've introduced and discussed power functions including notation involving fractional powers, which are really just nth roots in disguise and polynomial functions which are formed from power functions by adding up constant multiples of non-negative integer powers of our variable x. We described and illustrated important techniques of factorizing polynomials, in particular, the method of polynomial long division. Factorization is very important so it's well worth investing some time in mastering these techniques. Please read the notes that accompany this video and when you're ready please attempt the exercises. Thank you very much for watching and I look forward to seeing you again soon.
