Cambridge Additional Mathematics

164 Polynomials (Chapter 6)

Example 10 Self Tutor

Findallquartic polynomials with zeros 2 , ¡^13 , and ¡ 1 §

p

5.

The zeros ¡ 1 §

p

5 have sum=¡1+

p

5 ¡ 1 ¡

p

5=¡ 2 and

product=(¡1+

p

5)(¡ 1 ¡

p

5) =¡ 4

) they come from the quadratic factor x^2 +2x¡ 4.

The zeros 2 and¡^13 come from the linear factors x¡ 2 and 3 x+1.

) P(x)=a(x¡2)(3x+ 1)(x^2 +2x¡4), a 6 =0.

6 Findallquartic polynomials with zeros of:

a § 1 , §

p

2 b 2 , ¡^15 , §

p

3 c ¡ 3 ,^14 , 1 §

p

2 d 2 §

p

5 , ¡ 2 §

p

7

POLYNOMIAL EQUALITY

Two polynomials areequalif and only if they have thesame degree(order), and corresponding terms

have equal coefficients.

If we know that two polynomials areequalthen we canequate coefficientsto find unknown coefficients.

For example, if 2 x^3 +3x^2 ¡ 4 x+6=ax^3 +bx^2 +cx+d, where a,b,c,d 2 R, then

a=2, b=3, c=¡ 4 , and d=6.

Example 11 Self Tutor

Find constantsa,b, andcgiven that:

6 x^3 +7x^2 ¡ 19 x+7=(2x¡1)(ax^2 +bx+c) for allx.

6 x^3 +7x^2 ¡ 19 x+7=(2x¡1)(ax^2 +bx+c)

) 6 x^3 +7x^2 ¡ 19 x+7=2ax^3 +2bx^2 +2cx¡ax^2 ¡bx¡c

) 6 x^3 +7x^2 ¡ 19 x+7=2ax^3 +(2b¡a)x^2 +(2c¡b)x¡c

Since this is true for allx, we equate coefficients:

) 2 a=6

x^3 s

2 b¡a=7

x^2 s

2 c¡b=¡ 19

xs

and 7=¡c

constants

) a=3 and c=¡ 7 and consequently 2 b¡3=7 and ¡ 14 ¡b=¡ 19

) b=5

in both equations

So, a=3, b=5, and c=¡ 7.

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_06\164CamAdd_06.cdr Friday, 20 December 2013 12:59:47 PM BRIAN