Test your understanding of the Remainder Theorem.
1
Find the remainder when x² - 2x + 4 is divided by x - 2.
Explanation:
Using Remainder Theorem, p(2) = (2)² - 2(2) + 4 = 4 - 4 + 4 =
4.
2
If p(x) is divided by x + 1, the remainder is:
-
A
p(1)
-
B
p(-1)
-
C
p(0)
-
D
-p(1)
Explanation:
Comparing x + 1 with x - a, we get a = -1. So remainder is
p(-1).
3
What is the remainder when x³ + 3x² + 3x + 1 is divided by x + 1?
Explanation:
p(-1) = (-1)³ + 3(-1)² + 3(-1) + 1 = -1 + 3 - 3 + 1 = 0.
4
If p(x) = x³ - ax² + 6x - a is divided by x - a, remainder is:
Explanation:
p(a) = a³ - a(a)² + 6a - a = a³ - a³ + 5a = 5a.
5
The remainder when 4x³ - 12x² + 14x - 3 is divided by 2x - 1 is:
Explanation:
Divisor 2x - 1 = 0 → x = 1/2.
p(1/2) = 4(1/8) - 12(1/4) + 14(1/2) - 3
= 1/2 - 3 + 7 - 3 = 1/2 + 1 = 3/2.
6
When p(x) is divided by x, the remainder is:
Explanation:
Checking x = 0 (since x - 0 = x), the remainder is p(0).
7
Check whether the polynomial p(x) = 4x³ + 4x² - x - 1 is a multiple of 2x + 1.
-
A
Yes
-
B
No
-
C
Only for positive x
-
D
None
Explanation:
Zero of 2x + 1 is -1/2.
p(-1/2) = 4(-1/8) + 4(1/4) - (-1/2) - 1
= -1/2 + 1 + 1/2 - 1 = 0.
Since remainder is 0, it is a multiple.
8
If x - 1 divides the polynomial kx² - 3x + k completely, find k.
Explanation:
Since it divides completely, remainder p(1) = 0.
k(1)² - 3(1) + k = 0
2k - 3 = 0 → k = 3/2.
9
If p(x) = x⁹⁹ + 1 is divided by x + 1, remainder is:
Explanation:
p(-1) = (-1)⁹⁹ + 1 = -1 + 1 = 0.
10
What is the remainder when x⁴ + 1 is divided by x - 1?
Explanation:
p(1) = (1)⁴ + 1 = 1 + 1 = 2.
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