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4.6
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The power of x

Powers of x and higher polynomials

There is a family of relations that includes linear, quadratic and inverse relations all in the same framework. This is the family of functions defined by a power of \(\normalsize{x}\).

In this step we look at different powers of \(\normalsize{x}\), and see how they are the building blocks of polynomials.

Linear, quadratic and cubic powers

We have had a good look at linear powers, such as \(\normalsize{y=x}\), quadratic powers such as \(\normalsize{y=x^2}\) and cubic powers such as \(\normalsize{y=x^3}\) . Here are their graphs:

Graphs of y=x, y=x^2 and y=x^3 in the range [-12,14]

It is important to understand how higher powers of \(\normalsize{x}\) behave. The general pattern follows the three functions above, but generally as the exponent increases, the function becomes more extreme — and it gets very big more quickly as \(\normalsize{x}\) increases.

Higher powers of \(\normalsize{x}\)

Recall that \(\normalsize{x^0=1}\) for any \(\normalsize{x}\). So the function \(\normalsize{y=x^0}\) is the same as the function \(\normalsize{y=1}\) — that is, it is a constant function that does not change at all. In the following diagram, you can see the functions \(\normalsize{y=x^n}\) for \(\normalsize{n=0,1,2,3,4,5}\) and how they grow with \(\normalsize{x}\).

Plots of x^n for n=0,1,2,3,4,5. All curves go through the origin. Some have all positive values and some have both positive and negative values. Some increase more quickly than others.

Q1 (E): In the above graph, which function is which?

When the degree \(\normalsize{n}\) of the function \(\normalsize{y=x^n}\) is an odd natural number, such a polynomial always increases as \(\normalsize{x}\) increases. In this case we say that \(\normalsize{y=x^n}\) is an increasing function. However if \(\normalsize{n}\) is even, then as \(\normalsize{x}\) increases, two things can happen: the function decreases while \(\normalsize{x}\) is negative, and then increases for positive \(\normalsize{x}\).

Index laws

When working with powers it is useful to remember the index laws

\[\Large{x^r x^s =x^{r+s}}\] \[\Large{(x^r)^s =x^{rs}}\] \[\Large{(xy)^r =x^r y^r}\]

and for good measure we also include the rule that lets us work with negative exponents:

\[\Large{x^{-r}=\frac{1}{x^r}}.\]

We will shortly be talking about powers with fractional exponents. It will then be important to remember that these index laws still hold.

Growth rates of powers

The differences between various powers often becomes more noticeable for very small or very large values of \(\normalsize{x}\). While there is not so much difference between \(\normalsize{(1.5)^2}\) and \(\normalsize{(1.5)^3}\), there is quite a big difference between \(\normalsize{(150)^2}\) and \(\normalsize{(150)^3}\), and also between \(\normalsize{(0.015)^2}\) and \(\normalsize{(0.015)^3}\).

We say that the power \(\normalsize{y=x^3}\) has a larger growth rate than \(\normalsize{y=x^2}\). This becomes ever more noticeable as the exponent \(\normalsize{n}\) in \(\normalsize{y=x^n}\) increases.

Q2 (E): Which has a faster growth rate: the function \(7x^5\) or \(5x^7\)?

Q3 (M): What about when \(\normalsize x\) is getting closer to zero? Which of the functions \(7x^5\) and \(5x^7\) will decrease faster?

Combining powers of \(x\)

The simplest polynomials are the power functions \(\normalsize{y=x^n}\) for natural numbers \(\normalsize{n}\), and they form the basic building blocks to make more general polynomials, such as \(\normalsize{y=3x^5-4x^4+2x-7}\).

These are an important class of objects, because they are exactly the natural domain of algebra: we can add, subtract and multiply polynomials in much the same way as we do arithmetic with numbers, and sometimes we can also divide them. But polynomials are considerably more complicated than numbers, especially as we increase the degree.

Nevertheless, the basic guiding principle — that polynomials give us a domain of arithmetic that extends that of ordinary numbers — is a powerful and useful one.

Polynomials as products of factors

There is another way of getting polynomials rather than combining multiples of powers of \(\normalsize x\). Just as one way of getting bigger numbers is to multiply them together (for example \(\normalsize{1356= 2^23\times113}\)) , so too for polynomials we can multiply smaller polynomials together to get higher degree polynomials. The simplest way of doing that is to multiply together linear factors, for example

\[\Large{\begin{array}{rll} y &=p(x)=(x+7)(x+3)(x-2)(x+4) \\ &= x^4+12x^3+33x^2-38x-168\end{array}}\]

By Descartes’ theorem (Factors of quadratic polynomials and zeroes) each linear factor of \(\normalsize p\) corresponds to a zero of the function, so for example in this case \(\normalsize{p(-7)=p(-3)=p(2)=p(-4)=0}\).

You should be aware that a polynomial that has been composed as a product of linear factors is arithmetically rather special. In general a polynomial of degree \(\normalsize n\) may have less than \(\normalsize n\) linear factors. For example \(\normalsize x^2 + 1\) has no linear factors.

Shapes of polynomials functions

As we go up in degree, polynomial functions get longer to write down, with more coefficients, and their graphs become more complicated. Nevertheless there is a kind of predictability about the overall shape of polynomials that it is essential for us to understand. Even a higher degree polynomial function such as

\[\Large{y=\frac{3}{100} x(x + 4) (x + 1) (x - 3) (x - 4)}\]

has something of a regular graph: as we view it from left to right, it goes up, then down, then up, then down and then finally up.

Graph of the curve described above, with a line intersecting this curve at 5 points

The number of ups and downs is limited by the degree. A fundamental fact about polynomials is that the graph of a degree n polynomial can meet an arbitrary line in at most n points. This is an important theorem. Another variant is that a polynomial of degree \(\normalsize n\) can change direction — from up to down, or from down to up, as we view it from left to right — at most \(\normalsize (n-1)\) times.

But remember that not all polynomial functions exhibit this up, down, up, down aspect explicitly. In particular the powers of \(\normalsize{y=x^n}\) that we began our discussion with only go up if \(\normalsize{n}\) is odd, and go down and then up if \(\normalsize{n}\) is even, no matter how high the degree \(\normalsize n\).

Answers

A1. The function \(\normalsize y=x^0=1\) is light blue (the horizontal line), \(\normalsize y=x^1=x\) is red (the straight line going through the point \(\normalsize{[0.5, 0.5]}\)), \(\normalsize y=x^2\) is yellow (one of the 2 ‘even’ functions — with all positive values), \(\normalsize y=x^3\) is orange (one of the 3 ‘odd’ functions — with negative values for x < 0), \(\normalsize y=x^4\) is green (again an even function) and \(\normalsize y=x^5\) is dark blue (again an odd function).

A2. Since \(\normalsize 7\) is greater than \(\normalsize 5\) the function \(\normalsize x^7\) grows faster than \(\normalsize x^5\) as \(\normalsize x\) gets larger. The coefficients are relatively unimportant, so that also \(\normalsize 5x^7\) grows faster than \(\normalsize 7x^5\).

A3. Since \(\normalsize 7\) is greater than \(\normalsize 5\) the function \(\normalsize x^7\) decreases faster than \(\normalsize x^5\) as \(\normalsize x\) gets smaller. So also \(\normalsize 5x^7\) decreases faster than \(\normalsize 7x^5\) as \(\normalsize x\) gets smaller.

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Maths for Humans: Linear, Quadratic & Inverse Relations

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