Factor Theorem :
Let P(x) be a polynomial.
If (x - k) be a factor of P(x), then
P(k) = 0
Note :
If P(k) = 0, then (x - k) is not a factor of P(x).
Problem 1 :
Find the value of k, if (x - 3) is a factor of
f(x) = x3 - 11x + k
Solution :
Given : (x - 3) is a factor of f(x).
By Factor Theorem,
f(3) = 0
33 - 11(3) + k = 0
27 - 33 + k = 0
-6 + k = 0
k = 6
Problem 2 :
Find the value of k, if (x + 1) is a factor of
f(x) = kx5 - 121x3 - 15x2 - 25
Solution :
Given : (x + 1) is a factor of f(x).
By Factor Theorem,
f(-1) = 0
k(-1)5 - 121(-1)3 - 15(-1)2 - 25 = 0
k(-1) - 121(-1) - 15(1) - 25 = 0
-k + 121 - 15 - 25 = 0
-k + 81 = 0
k = 81
Problem 3 :
What is the value of k, if (x + 2) is a factor of
f(x) = -(x3 + 3x2) - 4(x - a)
Solution :
Given : (x + 2) is a factor of f(x).
By Factor Theorem,
f(-2) = 0
-[(-2)3 + 3(-2)2] - 4(-2 - a) = 0
-[-8 + 3(4)] + 8 + 4a = 0
-[-8 + 12] + 8 + 4a = 0
-4 + 8 + 4a = 0
4 + 4a = 0
4a = -4
a = -1
Problem 4 :
If (x - 2) is a factor of polynomial
P(x) = a(x3 - 2x) + b(x2 - 5),
which of the following must be true?
(A) a + b = 0
(B) 2a - b = 0
(C) 2a + b = 0
(D) 4a - b = 0
Solution :
Given : (x - 2) is a factor of f(x).
By Factor Theorem,
f(2) = 0
a[23 - 2(2)] + b(22 - 5) = 0
a(8 - 4) + b(4 - 5) = 0
a(4) + b(-1) = 0
4a - b = 0
Kindly mail your feedback to v4formath@gmail.com
We always appreciate your feedback.
©All rights reserved. onlinemath4all.com
Oct 03, 23 12:56 AM
Oct 03, 23 12:34 AM
Oct 02, 23 11:40 PM