Cambridge Additional Mathematics

Polynomials (Chapter 6) 165

Example 12 Self Tutor

Find constantsaandbif z^4 +9=(z^2 +az+ 3)(z^2 +bz+3) for allz.

z^4 +9=(z^2 +az+ 3)(z^2 +bz+3) for allz

) z^4 +9=z^4 +bz^3 +3z^2

+az^3 +abz^2 +3az

+3z^2 +3bz+9

) z^4 +9=z^4 +(a+b)z^3 +(ab+6)z^2 +(3a+3b)z+9 for allz

Equating coefficients gives

a+b=0 .... (1) fz^3 sg

ab+6=0 .... (2) fz^2 sg

3 a+3b=0 .... (3) fzsg

From (1) and (3) we see that b=¡a

) in (2), a(¡a)+6=0

) a^2 =6

) a=§

p

6 and so b= ̈

p

6

) a=

p

6 , b=¡

p

6 or a=¡

p

6 , b=

p

6

EXERCISE 6B.2

1 Find constantsa,b, andcgiven that:

a 2 x^2 +4x+5=ax^2 +[2b¡6]x+c for allx

b 2 x^3 ¡x^2 +6=(x¡1)^2 (2x+a)+bx+c for allx

c 6 x^3 ¡ 13 x^2 +7x+4=(3x+ 1)(ax^2 +bx+c) for allx.

2 Find constantsaandbif:

a z^4 +4=(z^2 +az+ 2)(z^2 +bz+2) for allz

b 2 z^4 +5z^3 +4z^2 +7z+6=(z^2 +az+ 2)(2z^2 +bz+3) for allz.

3aGiven that x^3 +9x^2 +11x¡21 = (x+ 3)(ax^2 +bx+c), find the values ofa,b, andc.

b Hence, fully factorise x^3 +9x^2 +11x¡ 21.

4aGiven that 4 x^3 +12x^2 +3x¡5=(2x¡1)(px^2 +qx+r), find the values ofp,q, andr.

b Hence, find the solutions to 4 x^3 +12x^2 +3x¡5=0.

5aGiven that 3 x^3 +10x^2 ¡ 7 x+4=(x+ 4)(ax^2 +bx+c), find the values ofa,b, andc.

b Hence, show that 3 x^3 +10x^2 ¡ 7 x+4 has only one real zero.

6 Suppose 3 x^3 +kx^2 ¡ 7 x¡2=(3x+ 2)(ax^2 +bx+c).

a Find the values ofa,b,c, andk.

b Hence, find the roots of 3 x^3 +kx^2 ¡ 7 x¡2=0.

7aFind real numbersaandbsuch that x^4 ¡ 4 x^2 +8x¡4=(x^2 +ax+ 2)(x^2 +bx¡2).

b Hence, find the real roots of x^4 +8x=4x^2 +4.

When simultaneously

solving more equations

than there are unknowns,

we must check that any

solutions fit equations.

If they do not, there are

.

all

no solutions

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Y:\HAESE\CAM4037\CamAdd_06\165CamAdd_06.cdr Friday, 20 December 2013 1:20:05 PM BRIAN