CIE Centre A-level Further Pure Maths Polynomials CIE Centre A-level Further Pure Maths © Adam Gibson

Recall the basic structure of the cubic polynomial. If Polynomials Again Recall the basic structure of the cubic polynomial. If and: Then: Sum of the roots = Note why we have divided through by a. Note also the signs (+ or -) for each term. For the 4th order example given, ask students to give values of the ak coefficients The graph is a good subject for discussion of the nature of higher order polynomials Don’t forget to ask the students what a 1st order polynomial is. Product of the roots = Pairwise product of the roots =

Let’s put it in a broader context: Polynomials Again Let’s put it in a broader context: For the 4th order example given, ask students to give values of the ak coefficients The graph is a good subject for discussion of the nature of higher order polynomials Don’t forget to ask the students what a 1st order polynomial is. We can always set the leading coefficient to 1 by dividing through by a. Q: What is the pattern in the second coefficient (b)? Q: What is the pattern in the trailing coefficient?

The coefficient of xn-1 is always -1 x the sum of the roots Polynomials Again Answers: The coefficient of xn-1 is always -1 x the sum of the roots (assuming you have made the leading coefficient 1). The constant term, or trailing coefficient = (-1)n x the product of the roots. The other coefficients can also be expressed in terms of the roots, but their form is obviously more complicated. For the 4th order example given, ask students to give values of the ak coefficients The graph is a good subject for discussion of the nature of higher order polynomials Don’t forget to ask the students what a 1st order polynomial is. Now let’s start to look at how we can manipulate polynomials to find interesting results.

Now write down the required polynomial. Polynomials Again The roots of are denoted as: Write down a polynomial with roots: Think … For the 4th order example given, ask students to give values of the ak coefficients The graph is a good subject for discussion of the nature of higher order polynomials Don’t forget to ask the students what a 1st order polynomial is. Now write down the required polynomial.

Polynomials Again For the 4th order example given, ask students to give values of the ak coefficients The graph is a good subject for discussion of the nature of higher order polynomials Don’t forget to ask the students what a 1st order polynomial is.

Now we will look at a more difficult problem Polynomials Again We solved this problem using substitution. The method can be applied where the new roots are related to the old roots through an invertible function: For the 4th order example given, ask students to give values of the ak coefficients The graph is a good subject for discussion of the nature of higher order polynomials Don’t forget to ask the students what a 1st order polynomial is. Now we will look at a more difficult problem involving the roots of a cubic…

Investigate!  Polynomials Again The roots of are denoted as: Find the value of: Investigate!  For the 4th order example given, ask students to give values of the ak coefficients The graph is a good subject for discussion of the nature of higher order polynomials Don’t forget to ask the students what a 1st order polynomial is.

Factorization – not possible Polynomials Again Factorization – not possible Sketching the graph might help From this we can only see that there is one real root > 2; but we need exact values. For the 4th order example given, ask students to give values of the ak coefficients The graph is a good subject for discussion of the nature of higher order polynomials Don’t forget to ask the students what a 1st order polynomial is. This curious question can actually be answered by deducing a recurrence relation for the sum of the powers of the roots. Let’s see how it works.

for each root. So we can write three equations: Polynomials Again We know that: for each root. So we can write three equations: 1 X For the 4th order example given, ask students to give values of the ak coefficients The graph is a good subject for discussion of the nature of higher order polynomials Don’t forget to ask the students what a 1st order polynomial is. 2 X 3 X

So by adding the equations together vertically we get: Polynomials Again So by adding the equations together vertically we get: The terms in {} all have the same structure. Let’s call this “sum of powers of roots” Sn, i.e.: For the 4th order example given, ask students to give values of the ak coefficients The graph is a good subject for discussion of the nature of higher order polynomials Don’t forget to ask the students what a 1st order polynomial is. Rewriting the equation using Sn gives the more compact form on the next slide:

We therefore have a “recurrence relation” defining Polynomials Again Q: What is S0 ? A: 3. We therefore have a “recurrence relation” defining an inductive sequence. But to find S7 it is not enough to know S0! For the 4th order example given, ask students to give values of the ak coefficients The graph is a good subject for discussion of the nature of higher order polynomials Don’t forget to ask the students what a 1st order polynomial is. Q: What are S1 and S2 ?

S1 (and slightly less easily, S2) can be calculated from Polynomials Again S1 (and slightly less easily, S2) can be calculated from the previous equations: For the 4th order example given, ask students to give values of the ak coefficients The graph is a good subject for discussion of the nature of higher order polynomials Don’t forget to ask the students what a 1st order polynomial is. Thus we can find S1 and S2 just by examining the coefficients of the cubic equation. All other Sns can be deduced (although it might be quite tedious!).

You should now be able to do this problem… Polynomials Again The roots of are denoted as: Find the value of: You should now be able to do this problem… For the 4th order example given, ask students to give values of the ak coefficients The graph is a good subject for discussion of the nature of higher order polynomials Don’t forget to ask the students what a 1st order polynomial is.

To go further and find S7, you must write down Worked Solution For the 4th order example given, ask students to give values of the ak coefficients The graph is a good subject for discussion of the nature of higher order polynomials Don’t forget to ask the students what a 1st order polynomial is. To go further and find S7, you must write down the recurrence relation for Sn.

The remainder is trivial, if laborious, calculation: Worked Solution Do you see how to get this? The remainder is trivial, if laborious, calculation: For the 4th order example given, ask students to give values of the ak coefficients The graph is a good subject for discussion of the nature of higher order polynomials Don’t forget to ask the students what a 1st order polynomial is.

…which finally gives the desired result: Worked Solution …which finally gives the desired result: This example is rather complex, but only a little more difficult than the questions you will find on the exam. For the 4th order example given, ask students to give values of the ak coefficients The graph is a good subject for discussion of the nature of higher order polynomials Don’t forget to ask the students what a 1st order polynomial is.

Here are the key points so far: Review Here are the key points so far: Remember the relationship between the coefficients and the roots of a cubic polynomial Remember what you learnt in P3 about real/complex roots, curve sketching and factorizing using the factor theorem Realize that it is possible to find relationships between the sum of the powers of the roots (recurrence relationships, specifically) Realize that sometimes by making a substitution you can find further results about the roots For the 4th order example given, ask students to give values of the ak coefficients The graph is a good subject for discussion of the nature of higher order polynomials Don’t forget to ask the students what a 1st order polynomial is.

PROBLEMS A cubic polynomial has root +2. The product of its three roots is -9/2. The sum of its roots is 43/4. Write down the polynomial. How many different cubic polynomials with positive integer roots have -98 as their constant term? (Assume the coefficient of x3 is 1). If that was easy, try this one: how many different polynomials of degree 7, with positive integer roots, have -9800 as their constant term? Can you find a generalized algorithm to solve this kind of problem? For the 4th order example given, ask students to give values of the ak coefficients The graph is a good subject for discussion of the nature of higher order polynomials Don’t forget to ask the students what a 1st order polynomial is. Answers: 1) (x-2)(x-9)(x+1/4) 2) 5 3) 65 General formula is , where n is the number of prime factors.

Polynomials – tips and tricks The most useful ideas to remember: Note that this is always true, for polynomials of any degree. For example, degree 4: For the 4th order example given, ask students to give values of the ak coefficients The graph is a good subject for discussion of the nature of higher order polynomials Don’t forget to ask the students what a 1st order polynomial is. Can you see why this must be true?

Polynomials – tips and tricks For degree n: To see this, consider degree 3 (note the reason for the negative sign!): For the 4th order example given, ask students to give values of the ak coefficients The graph is a good subject for discussion of the nature of higher order polynomials Don’t forget to ask the students what a 1st order polynomial is.

Polynomials – tips and tricks Having these two simple formulas for S(2) and S(-1), combined with the fact that S(1) and S(0) can be observed immediately, makes it easy to find any S(n). For the 4th order example given, ask students to give values of the ak coefficients The graph is a good subject for discussion of the nature of higher order polynomials Don’t forget to ask the students what a 1st order polynomial is.