Question

if the polynomial f(x)=2x^3+mx^2+nx-14 has (x-1) and (x-2) as its factors find the value of m/n

Answer 1

hey your ans is here
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given
(x-1);(x-2) as factors
so
1)x-1=0
x=1
putting values in 2x3+mx2+nx-14=0
2×(1)3+m×(1)2+n×(1)-14=0
2+m+n-14=0
m+n-12=0
m+n=12------------ (1)

2)x-2=0
x=2
putting values
2×(2)3+m(2)2+n(2)-14=0
16+4m+2n-14=0
4m+2n+2=0
2(2m+n+1)=0
2m+n+1=0
2m+n=-1-----------(2)

now through substitution
m+n=12
2m+n=-1

substituting the values
n=12-m
2m+12-m=-1
m=-13

-13+n=12
n=25
so
m/n=-13/25
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hope it helps!

Answer 2

Answer:

We have to find the relation between variables given in the problem. Let us proceed to that.

Step-by-step explanation:

Given: The polynomial = 2x³+mx²+nx-14, where m and n are variables, and the factors of this polynomial as (x-1) and (x-2).

To Find: The value of  $\frac{m}{n}$.

Now, as (x-1) and (x-2) are factors of this polynomial, x = 1 and 2 are the solutions of the polynomial. Therefore, putting these values in the equation will equate them to Zero.

Hence, it will become :

2(1)³ + m(1)² + n(1) - 14 = 0        →          2+m+n-14 = 0

             ⇒ m+n = 12                                    (eq. 1)

Similarly, 2(2)³ + m(2)² + n(2) - 14 = 0       →       16+4m+2n-14 = 0

             ⇒ 4m+2n = -2      

             ⇒ 2m+n = -1                                  (eq. 1)

Hence, eliminating Eq. 1 and 2, we get :

                    ⇒ m = -13 and n = 25

Hence, the relation $\frac{m}{n}$ would be equal to -$\frac{-13}{25}$