Cambridge Additional Mathematics

166 Polynomials (Chapter 6)

Example 13 Self Tutor

x+3is a factor of P(x)=x^3 +ax^2 ¡ 7 x+6. Find a 2 R and the other factors.

Since x+3is a factor, The coefficient of

x^3 is 1 £1=1

This must be 2 so the

constant term is 3 £2=6

x^3 +ax^2 ¡ 7 x+6=(x+ 3)(x^2 +bx+2) for some constantb

=x^3 +bx^2 +2x

+3x^2 +3bx+6

=x^3 +(b+3)x^2 +(3b+2)x+6

Equating coefficients gives 3 b+2=¡ 7 and a=b+3

) b=¡ 3 and a=0

) P(x)=(x+ 3)(x^2 ¡ 3 x+2)

=(x+ 3)(x¡1)(x¡2)

The other factors are (x¡1) and (x¡2).

8

Example 14 Self Tutor

2 x+3and x¡ 1 are factors of 2 x^4 +ax^3 ¡ 3 x^2 +bx+3.

Find constantsaandband all zeros of the polynomial.

Since 2 x+3and x¡ 1 are factors, The coefficient ofx^4

is 2 £ 1 £1=2

This must be¡ 1 so the constant

term is 3 £¡ 1 £¡1=3

2 x^4 +ax^3 ¡ 3 x^2 +bx+3=(2x+ 3)(x¡1)(x^2 +cx¡1) for somec

=(2x^2 +x¡3)(x^2 +cx¡1)

=2x^4 +2cx^3 ¡ 2 x^2

+ x^3 +cx^2 ¡ x

¡ 3 x^2 ¡ 3 cx+3

=2x^4 +(2c+1)x^3 +(c¡5)x^2 +(¡ 1 ¡ 3 c)x+3

Equating coefficients gives 2 c+1=a, c¡5=¡ 3 , and ¡ 1 ¡ 3 c=b

) c=2

) a=5and b=¡ 7

) P(x)=(2x+ 3)(x¡1)(x^2 +2x¡1)

Now x^2 +2x¡ 1 has zeros

¡ 2 §

p

4 ¡4(1)(¡1)

2

=¡^2 §^2

p

2

2

=¡ 1 §

p

2

) P(x) has zeros ¡^32 , 1 , and ¡ 1 §

p

2.

2 x¡ 3 is a factor of 2 x^3 +3x^2 +ax+3. Find a 2 R and all zeros of the cubic.

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_06\166CamAdd_06.cdr Tuesday, 21 January 2014 2:36:44 PM BRIAN