## Chapter 2 Polynomials Exercise 2.3

Question 1: Find the remainder when x³ + 3x² + 3x + 1 is divided by

i) x + 1

ii) x - 1/2

iii) x

iv) x + π

v) 5 + 2x

Answer:

i)

Using remainder theorem if p(x) is divided by (x - a) the remainder is p(a).

The zero of x + 1 is -1.

p(-1) = (-1)³ + 3(-1)² + 3(-1) + 1

= -1 + 3 - 3 + 1

= 0

Using long division method

ii)

Using remainder theorem if p(x) is divided by (x - a) the remainder is p(a).

The zero of x - 1/2 is 1/2.

p(1/2) = (1/2)³ + 3(1/2)² + 3(1/2) + 1

= 1/8 + 3/4 + 3/2 + 1

= 1 + 6 + 12 + 8/8

= 27/8

Using long division

iii)

Using remainder theorem if p(x) is divided by (x - a) the remainder is p(a).

The zero of x is 0.

p(0) = (0)³ + 3(0)² + 3(0) + 1

= 0 + 0 + 0 + 1

= 1

Using long division

iv)

Using remainder theorem if p(x) is divided by (x - a) the remainder is p(a).

The zero of x + π is -π.

p(-π) = (-π)³ + 3(-π)² + 3(-π) + 1

= -π³ + 3π² - 3π + 1

Using long division

v)

Using remainder theorem if p(x) is divided by (x - a) the remainder is p(a).

The zero of 5 + 2x is -5/2.

p(-5/2) = (-5/2)³ + 3(-5/2)² + 3(-5/2) + 1

= -125/8 + 75/4 + -15/2 + 1

= -27/8

Using long division

Question 2: Find the remainder when x³ - ax² + 6x - a is divided by x - a.

Answer: Using remainder theorem if p(x) is divided by (x - a) the remainder is p(a).

The zero of x - a is a.

p(a) = x³ - ax² + 6x - a

= (a)³ - a(a)² + 6(a) - a

= a³ - a³ + 6a - a

= 5a

Question 3: Check whether 7 + 3x is a factor of 3x³ + 7x.

Answer: Using remainder theorem if p(x) is divided by (x - a) the remainder is p(a).

The zero of 3x + 7 is -7/3.

p(-7/3) = 3(-7/3)³ + 7(-7/3)

= 3(-343/27) - 49/3

= -343 - 147/9

= -490/9

Therefore, 7 + 3x is not a factor of 3x³ + 7x.

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