find the values of p and q if x²-1 is a factor of x⁴+px³+2x²-3x+q (ans:p=3,q=-3)
Answers
Answer :
p = 3
q = -3
Step-by-step explanation :
Given :
x² - 1 is a factor of x⁴ + px³ + 2x² - 3x + q
To find :
the values of p and q
Solution :
Since x² - 1 is a factor,
➙ x² - 1 = 0
➙ x² = 1
➙ x = √1
➙ x = +1 , -1
+1 and -1 are the zeroes of the given polynomial.
Put x = +1,
Since it's a zero, when we substitute x = 1, the result is zero.
(1)⁴ + p(1)³ + 2(1)² - 3(1) + q = 0
1 + p(1) + 2(1) - 3 + q = 0
1 + p + 2 - 3 + q = 0
p + q + 3 - 3 = 0
p + q = 0 ➙ [1]
Put x = -1,
Since it's also a zero, when we substitute x = -1, the result is zero.
(-1)⁴ + p(-1)³ + 2(-1)² - 3(-1) + q = 0
1 + p(-1) + 2(1) + 3 + q = 0
1 - p + 2 + 3 + q = 0
-p + q + 6 = 0
-p + q = -6 ➙ [2]
Adding both the equations,
p + q - p + q = 0 - 6
2q = -6
q = -6/2
q = -3
Substitute in equation [1],
p + q = 0
p - 3 = 0
p = 3
Therefore, p = 3 and q = -3