If x+2and x-1 are the factors of x³+10x²+mx+n, then the value of m and n are respectively
A. 5 and -3
B. 17 and -8
C. 7 and -18
D. 23 and -19
Answers
Given : If (x + 2) and (x - 1) are the factors of x³ + 10x² + mx + n
To find : The value of m and n
Let p(x) = x³ + 10x² + mx + n be the given polynomial. If (x + 2) and (x - 1) are factors of p(x) , then p(- 2) = 0 and p(1) = 0 :
⇒(-2)³ + 10 (-2)² + m (-2) + n = 0
⇒ - 8 + 10 × 4 - 2m + n = 0
⇒ - 8 + 40 - 2m + n = 0
⇒ 32 - 2m + n = 0 ………... (1)
Similarly p(1) = 0
⇒ (1)³ + 10 (1)² + m (1) + n = 0
⇒ 1 + 10 × 1 + m + n = 0
⇒ 1 + 10 + m + n = 0
⇒ 11 + m + n = 0 …………..(2)
On subtracting eq (1) from eq(2) :
⇒ 11 + m + n - ( 32 - 2m + n ) = 0
⇒ 11 + m + n - 32 + 2m - n = 0
⇒ 2m + m + n - n - 32 + 11 = 0
⇒ 3m - 21 = 0
⇒ 3m = 21
⇒ m = 21/3
⇒ m = 7
On putting the value of m = 7 in eq 2 :
11 + m + n = 0
⇒ 11 + 7 + n = 0
⇒ 18 + n = 0
⇒ n = - 18
Hence, the value of m and n is 7 & - 18.
Option (C) 7 and - 18 is correct.
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When x+2 and x-1 are factors of p(x)=x³+10x²+mx+n:
x+2=0
=> x=(-2)
p(-2)=0
=> (-2)³+10(-2)²+m(-2)+n=0
=> -8 + 40 - 2m + n = 0
=> n = 2m - 32 __(1)
x-1=0
=> x=1
p(1)=0
=> (1)³+10(1)²+m(1)+n=0
=> 1 + 10 + m + n = 0
=> 11 + m + n = 0 __(2)
Put (1) in (2):
=> 11 + m + 2m - 32 = 0
=> 3m = 21
=> m = 7
n=2m-32
=> n=14-32
=> n=(-18)
Option (c) 7 and -18 is the answer.