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Answers
Solution :
Given ,
n and m are the zeros of the polynomial F(x) = 3x^2 + 11x - 4
To Find
Value of m/n + n/m
Answer
⇒ 3 (m)^2 + 11 (m) - 4 = 0
⇒ 3m^2 + 11m - 4 = 0
⇒ 3m^2 + 12m - m -4 = 0
⇒ 3m ( m + 4 ) - 1 ( m + 4) = 0
⇒ ( 3m - 1) ( m + 4) = 0
⇒ m = 1/3 or 4
So , we replace n in x place we get some value
m/n + n/m = 1 + 1
=> 2
Therefore , the value of m/n + n/m is 2
Polynomial given
Let f ( x ) be 3 x² + 11 x - 4
What is a zero ?
A zero is the value of x for which the function f ( x ) is simple equal to zero .
There are many ways to approach the answer .. Seeing your copy I guess you need the first method .
Method 1 :
3 x² + 11 x - 4 = 0
Comparing with a x² + b x + c = 0
a = 3
b = 11
c = -4
Sum of roots = - b / a
Product of roots = c / a
Given that m and n are the zeroes .
m + n = - b / a
= > m + n = - 11 / 3 .............................(1)
= > m n = - 4 / 3..............................(2)
Solving the problem :
m / n + n / m
L.C.M = m × n
= > ( m² + n² ) / ( m n )
Now note that :
m² + n² = m² + n² + 2 m n - 2 m n = ( m + n )² - 2 m n
Hence :
( m² + n² ) / m n
= > [ ( m + n )² - 2 m n ] / m n
Now put the values of ( 1 ) and ( 2 ) :
= > [ ( - 11 / 3 )² - 2 ( - 4 / 3 ) ] / ( - 4 / 3 )
= > [ 121 / 9 + 8 / 3 ] / ( - 4 / 3 )
= > [ ( 121 + 8 × 3 ) / 9 ] / ( - 4 / 3 )
= > [ 121 + 24 ] / ( - 4 / 3 ) × 9
= > [ 145 ] / ( - 12 )
= > - 145 / 12
Method 2 :
3 x² + 11 x - 4 = 0
= > 3 x² + 12 x - x - 4 = 0
= > 3 x ( x + 4 ) - 1 ( x + 4 ) = 0
= > ( 3 x - 1 )( x + 4 ) = 0
Either
= > 3 x - 1 = 0
= > 3 x = 1
= > x = 1 / 3
Or :
= > x + 4 = 0
= > x = - 4
Let m be - 4 and n = 1 / 3
m / n + n / m
= > ( - 4 ) / ( 1 / 3 ) + ( 1 / 3 ) / ( - 4 )
= > - 12 - 1 / ( 12 )
= > ( - 144 - 1 ) / 12
= > - 145 / 12
ANSWER :
The value of m/n + n/m will be - 145 / 12