Further Pure 1 Chapter 2 :: �Roots of Polynomials
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Introduction
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The purpose of this chapter is to understand the underlying relationship between the roots of a polynomial, and the coefficients of each term.
It’s the constant term in the polynomial!
This is not yet exactly surprising: At GCSE you found the factorisation of a quadratic by finding two numbers that added to give the middle coefficient and multiplied to give the last number; this in turn allowed you to find the roots...
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Introduction
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Does this generalise to higher-order polynomials?
We will see there are very similar relationship between roots of higher order polynomials, and their coefficients:
Polynomial | Sum of roots | Sum of possible products of pairs of roots | Sum of products of triples |
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Back to roots of quadratics…
Use your formulae for the sum and product of roots. The underlying point here is that we can find these quantities without needing the roots themselves.
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Further Example
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Test Your Understanding
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Exercise 2A
Pearson Further Pure Mathematics Book 1
Pages 30
Expressions related to the roots of a polynomial
Such identities can be extended to cubics and quartics:
(You can use these results without proof)
(We can see these cubes formulae don’t generalise nicely as we increase the order of the polynomial. For this reason you are not required to know the sum of cubes for quartics)
Example
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Exercise 2B
Pearson Further Pure Mathematics Book 1
Pages 34