Cambridge Additional Mathematics

Polynomials (Chapter 6) 157

Example 1 Self Tutor

If P(x)=x^3 ¡ 2 x^2 +3x¡ 5 and Q(x)=2x^3 +x^2 ¡ 11 , find:

a P(x)+Q(x) b P(x)¡Q(x)

a P(x)+Q(x)

= x^3 ¡ 2 x^2 +3x¡ 5

+2x^3 + x^2 ¡ 11

=3x^3 ¡ x^2 +3x¡ 16

b P(x)¡Q(x)

= x^3 ¡ 2 x^2 +3x¡ 5 ¡(2x^3 +x^2 ¡11)

= x^3 ¡ 2 x^2 +3x¡ 5

¡ 2 x^3 ¡ x^2 +11

= ¡x^3 ¡ 3 x^2 +3x+6

SCALAR MULTIPLICATION

Tomultiplya polynomial by ascalar(constant) we multiply each term by the scalar.

Example 2 Self Tutor

If P(x)=x^4 ¡ 2 x^3 +4x+7, find:

a 3 P(x) b ¡ 2 P(x)

a 3 P(x)

=3(x^4 ¡ 2 x^3 +4x+7)

=3x^4 ¡ 6 x^3 +12x+21

b ¡ 2 P(x)

=¡2(x^4 ¡ 2 x^3 +4x+7)

=¡ 2 x^4 +4x^3 ¡ 8 x¡ 14

POLYNOMIAL MULTIPLICATION

Tomultiplytwo polynomials, we multiply each term of the first polynomial by each

term of the second polynomial, and then collect like terms.

Example 3 Self Tutor

If P(x)=x^3 ¡ 2 x+4 and Q(x)=2x^2 +3x¡ 5 , find P(x)Q(x).

P(x)Q(x)=(x^3 ¡ 2 x+ 4)(2x^2 +3x¡5)

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

=2x^5 +3x^4 ¡ 5 x^3

¡ 4 x^3 ¡ 6 x^2 +10x

+8x^2 +12x¡ 20

=2x^5 +3x^4 ¡ 9 x^3 +2x^2 +22x¡ 20

Collecting ‘like’ terms is

made easier by writing

them one above the other.

It is a good idea to place

brackets around expressions

which are subtracted.

4037 Cambridge

cyan magenta yellow black Additional Mathematics

(^05255075950525507595)
100 100
(^05255075950525507595)
100 100
Y:\HAESE\CAM4037\CamAdd_06\157CamAdd_06.cdr Thursday, 3 April 2014 5:13:03 PM BRIAN