Cambridge Additional Mathematics

162 Polynomials (Chapter 6)

Azeroof a polynomial is a value of the variable which makes the polynomial equal to zero.

®is azeroof polynomial P(x) , P(®)=0.

Therootsof a polynomialequationare the solutions to the equation.

®is aroot(orsolution)ofP(x)=0, P(®)=0.

Therootsof P(x)=0are thezerosof P(x) and thex-intercepts of the graph of y=P(x).

Consider P(x)=x^3 +2x^2 ¡ 3 x¡ 10

) P(2) = 2^3 + 2(2)^2 ¡3(2)¡ 10

=8+8¡ 6 ¡ 10

=0

This tells us: ² 2 is a zero of x^3 +2x^2 ¡ 3 x¡ 10

² 2 is a root of x^3 +2x^2 ¡ 3 x¡10 = 0

² the graph of y=x^3 +2x^2 ¡ 3 x¡ 10 has thex-intercept 2.

If P(x)=(x+ 1)(2x¡1)(x+2), then (x+1), (2x¡1), and (x+2)are itslinear factors.

Likewise P(x)=(x+3)^2 (2x+3) has been factorised into 3 linear factors, one of which is repeated.

x¡® is afactorof the polynomial P(x) , there exists a polynomial Q(x)

such that P(x)=(x¡®)Q(x).

Example 7 Self Tutor

Find the zeros of:

a x^2 ¡ 6 x+2 b x^3 ¡ 5 x

a We wish to findxsuch that

x^2 ¡ 6 x+2=0

) x=

6 §

p

36 ¡4(1)(2)

2

) x=

6 §

p

28

2

) x=

6 § 2

p

7

2

) x=3§

p

7

The zeros are 3 ¡

p

7 and 3+

p

7.

b We wish to findxsuch that

x^3 ¡ 5 x=0

) x(x^2 ¡5) = 0

) x(x+

p

5)(x¡

p

5) = 0

) x=0or §

p

5

The zeros are ¡

p

5 , 0 , and

p

5.

B Zeros, roots, and factors

An equation has.

A polynomial has.

roots

zeros

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_06\162CamAdd_06.cdr Friday, 20 December 2013 12:59:42 PM BRIAN