Cambridge Additional Mathematics

156 Polynomials (Chapter 6)

Opening problem

To determine whether 7 is afactorof 56 , we divide 56 by 7. The result is exactly 8. Since there is no

remainder, 7 is a factor of 56.

Things to think about:

a Can we perform a similar test foralgebraicfactors? For example, how can we determine whether

x¡ 3 is a factor of x^3 ¡ 4 x^2 +2x+3?

b Given that x¡ 3 is a factor of x^3 ¡ 4 x^2 +2x+3, what does this tell us about the graph of

f(x)=x^3 ¡ 4 x^2 +2x+3?

Up to this point we have studied linear and quadratic functions at some depth, with perhaps occasional

reference to cubic functions. These are part of a larger family of functions called thepolynomials.

Apolynomial functionis a function of the form

P(x)=anxn+an¡ 1 xn¡^1 +::::+a 2 x^2 +a 1 x+a 0 , a 1 , ....,anconstant, an 6 =0.

We say that: x is thevariable

a 0 is theconstant term

an is theleading coefficientand is non-zero

ar is thecoefficient ofxr for r=0, 1 , 2 , ....,n

n is thedegreeof the polynomial, being the highest power of the variable.

Insummation notation, we write P(x)=

Pn

r=0

arxr,

which reads: “the sum from r=0ton,ofarxr”.

Areal polynomial P(x) is a polynomial for which ar 2 R, r=0, 1 , 2 , ....,n.

The low degree members of the polynomial family have special names, some of which you are already

familiar with. For these polynomials, we commonly write their coefficients as a,b,c, ....

Polynomial function Degree Name

ax+b, a 6 =0 1 linear

ax^2 +bx+c, a 6 =0 2 quadratic

ax^3 +bx^2 +cx+d, a 6 =0 3 cubic

ax^4 +bx^3 +cx^2 +dx+e, a 6 =0 4 quartic

ADDITION AND SUBTRACTION

Toaddorsubtracttwo polynomials, we collect ‘like’ terms.

A REAL POLYNOMIALS

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_06\156CamAdd_06.cdr Thursday, 3 April 2014 5:12:32 PM BRIAN