Cambridge Additional Mathematics

158 Polynomials (Chapter 6)

EXERCISE 6A.1

1 If P(x)=x^2 +2x+3 and Q(x)=4x^2 +5x+6, find in simplest form:

a 3 P(x) b P(x)+Q(x) c P(x)¡ 2 Q(x) d P(x)Q(x)

2 If f(x)=x^2 ¡x+2 and g(x)=x^3 ¡ 3 x+5, find in simplest form:

a f(x)+g(x) b g(x)¡f(x) c 2 f(x)+3g(x)

d g(x)+xf(x) e f(x)g(x) f [f(x)]^2

3 Expand and simplify:

a (x^2 ¡ 2 x+ 3)(2x+1) b (x¡1)^2 (x^2 +3x¡2) c (x+2)^3

d (2x^2 ¡x+3)^2 e (2x¡1)^4 f (3x¡2)^2 (2x+ 1)(x¡4)

4 Find the following products:

a (2x^2 ¡ 3 x+ 5)(3x¡1) b (4x^2 ¡x+ 2)(2x+5)

c (2x^2 +3x+ 2)(5¡x) d (x¡2)^2 (2x+1)

e (x^2 ¡ 3 x+ 2)(2x^2 +4x¡1) f (3x^2 ¡x+ 2)(5x^2 +2x¡3)

g (x^2 ¡x+3)^2 h (2x^2 +x¡4)^2

i (2x+5)^3 j (x^3 +x^2 ¡2)^2

Discussion

Suppose f(x) is a polynomial of degreem, and g(x) is a polynomial of degreen.

What is the degree of:

² f(x)+g(x) ² 5 f(x) ² [f(x)]^2 ² f(x)g(x)?

DIVISION OF POLYNOMIALS

The division of polynomials is only useful if we divide a polynomial of degreenby another of degreenor

less.

Division by linears

Consider (2x^2 +3x+ 4)(x+2)+7.

If we expand this expression we obtain (2x^2 +3x+ 4)(x+2)+7=2x^3 +7x^2 +10x+15.

Dividing both sides by (x+2), we obtain

2 x^3 +7x^2 +10x+15

x+2

=

(2x^2 +3x+ 4)(x+2)+7

x+2

=(2x

(^2) +3x+ 4)(x+2)
x+2
+^7
x+2
=2x^2 +3x+4+
7
x+2
where x+2 is the divisor,
2 x^2 +3x+4 is the quotient,
and 7 is the remainder.
The division of polynomials is not required for
the syllabus, but is useful for understanding
the Remainder and Factor theorems.
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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_06\158CamAdd_06.cdr Thursday, 3 April 2014 5:18:13 PM BRIAN