861The Italian mathematician Paolo Ruffini,born in 1765, is responsible forsynthetic division, also known asRuffini’s rule, a technique used for thedivision of polynomials that is coveredin this chapter.Ruffini was not merely a mathematicianbut also held a licence to practisemedicine. During the turbulent years ofthe French Revolution, Ruffini lost hischair of mathematics at the university ofModena by refusing to swear an oath tothe republic. Ruffini seemed unbotheredby this, indeed the fact that he could nolonger teach mathematics meant that hecould devote more time to his patients,who meant a lot to him. It also gave hima chance to do further mathematicalresearch.The project he was working onwas to prove that the quintic equation cannot be solved by radicals. BeforeRuffini, no other mathematician published the fact that it was not possible to solvethe quintic equation by radicals. For example, Lagrange in his paper Reflections on theresolution of algebraic equationssaid that he would return to this question, indicatingthat he still hoped to solve it by radicals. Unfortunately, although his work wascorrect, very few mathematicians appeared to care about this new finding. Hisarticle was never accepted by the mathematical community, and the theorem isnow credited to being solved by Abel.

4PolynomialsPaolo Ruffini

4 Polynomials87This chapter treats thistopic as if a calculator isnot available throughoutuntil the section on usinga calculator at the end.This was covered in Chapter 3.DegreeForm of polynomialName of function1Linear2Quadratic3Cubic4Quartic5Quintica x5bx4c x3dx2e xfa x4bx3c x2dxea x3b x2˛c xda x2bxca xbis a polynomial is of degree 5 or quintic function. The coefficientof the leading term is 2, and is the constant term.Values of a polynomialWe can evaluate a polynomial in two different ways. The first method is to substitute thevalue into the polynomial, term by term, as in the example below.7f1x22x53x27ExampleFind the value of when Substituting:4812124f122233122261224x2.f1x2x33x26x4The second method is to use what is known as a nested schemeThis is where the coefficients of the polynomial are entered into a table, and then thepolynomial can be evaluated, as shown in the example below.

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4.1Polynomial functionsPolynomials are expressions of the type Theseexpressions are known as polynomials only when all of the powers of xare positiveintegers (so no roots, or negative powers). The degreeof a polynomial is the highestpower of x(or whatever the variable is called). We are already familiar with some ofthese functions, and those with a small degree have special names:f1x2axnbxn1...pxc.

To see why this nested calculation scheme works, consider the polynomial2x3x2x5.Using the nested calculation scheme, evaluate the polynomialwhenx2.f1x22x44x35x8This needs to be here asthere is no term.xEach of these is then multipliedby to give the number

4 Polynomials88Example

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