Question 1 :
Check whether p(x) is a multiple of g(x) or not .
p(x) = x3 - 5x2 + 4x - 3 ; g(x) = x – 2
Solution :
In order to check if g(x) is a multiple of p(x), let us check if g(x) is a factor of p(x).
x - 2 = 0
x = 2
p(x) = x3 - 5x2 + 4x - 3
p(2) = 23 - 5(2)2 + 4(2) - 3
= 5 - 5(4) + 8 - 3
= 5 - 20 + 8 - 3
= 13 -23
= -10 ≠ 0
(x - 2) is a not a factor of p(x). Hence it not a multiple of p(x).
Question 2 :
By remainder theorem, find the remainder when, p(x) is divided by g(x) where,
(i) p(x) = x3 - 2x2 - 4x - 1 and g(x) = x + 1
Solution :
x + 1 = 0
x = -1
p(x) = x3 - 2x2 - 4x - 1
p(-1) = (-1)3 - 2(-1)2 - 4(-1) - 1
p(-1) = -1 - 2(1) + 4 - 1
= -1 - 2 + 4 - 1
p(-1) = 0
The remainder is 0.
(ii) p(x) = 4x3 - 12x2 + 14x - 3 and g(x) = 2x - 1
Solution :
2x - 1 = 0
x = 1/2
p(x) = 4x3 - 12x2 + 14x - 3
p(1/2) = 4(1/2)3 - 12(1/2)2 + 14(1/2) - 3
p(1/2) = (4/8) - (12/4) + 7 - 3
= (1/2) - 3 + 7 - 3
= (1/2) + 1
= 3/2
Hence the remainder is 3/2.
(iii) p(x) = x3 - 3x2 + 4x + 50 and g(x) = x - 3
Solution :
x - 3 = 0
x = 3
p(x) = x3 - 3x2 + 4x + 50
p(3) = 33 - 3(3)2 + 4(3) + 50
= 27 - 27 + 12 + 50
= 62
Hence the remainder is 62.
Question 3 :
Find the remainder when 3x3 - 4x2 + 7x - 5 is divided by (x+3).
Solution :
p(x) = 3x3 - 4x2 + 7x - 5
x + 3 = 0
x = -3
p(-3) = 3(-3)3 - 4(-3)2 + 7(-3) - 5
= 3(-27) - 4(9) - 21 - 5
= -81 - 36 - 21 - 5
= -143
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