Question

3. If the zeroes of the cubic polynomial x3 – 6x2 + 3x + 10 are of the form m, m + n and m + 2n for some real numbers m and n, find the value of m and n​

Answer 1

Answer:

• m = 5 , n= -3 or m= -1 ,n=3
• zeroes= -1,2,5

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Step-by-step explanation:

Given: x³-6x²+3x+10

zeroes are of the form = m,m+n and m+2n

To find:

a=?,b=?

ß all zeroes of polynomial.

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$sum \: of \: roots = \frac{ - coefficient \: of \: {x}^{2} }{coefficient \: of \: {x}^{3} } \\ = > m + 2n + m + m + n = \frac{ - ( - 6)}{1} \\ = > 3m + 3n = 6 \\ = > 3(m + n) = 6 \\ = > m + n = 2 \: \: \: \: \: \: ....(1) \\ \\ product \: of \: roots = \frac{constant}{coefficient \: of \: {x}^{3} } \\ = > (m + 2n)(m + n)m = \frac{ - 10}{1} \\ = >( m + n + n)(m + n)m = - 10 \\ = > (2 + n)(2)m = - 10 \: \: \: \: \: (from \:eq.1) \\ = > (2 + n)2m = - 10 \\ = > (2 + 2 - m)2m = - 10 \\ = > (4 - m)2m = - 10 \\ = > 4m - {m}^{2} = - 5 \\ = > {m}^{2} - 4m - 5 = 0$

By solving quadratic equation we get,

m=5,-1

Putting value of m in eq.(1) we get,

When

m=5 = 5+n=2=>n= -3

m= -1= -1+n=2=>n=3.

when m=5 ß n= -3, zeroes are 5, [5+(-3)]=2,[5+2(-3)]= -6

when m= -1 ß n= -1, zeroes are -1,(-1+3)=2, [-1+2(3)]=5

therefore, the value of m= 5 ß n= -3 or m= -1 ß n=3.

zeroes are -1,2,5.

hope it was clear!!!

Answer 2

Answer:

Step-by-step explanation:

Answer:

m = 5 , n= -3 or m= -1 ,n=3

zeroes= -1,2,5

______________________________________

Step-by-step explanation:

Given: x³-6x²+3x+10

zeroes are of the form = m,m+n and m+2n

To find:

a=?,b=?

ß all zeroes of polynomial.

_____________________________________

By solving quadratic equation we get,

m=5,-1

Putting value of m in eq.(1) we get,

When

m=5 = 5+n=2=>n= -3

m= -1= -1+n=2=>n=3.

when m=5 ß n= -3, zeroes are 5, [5+(-3)]=2,[5+2(-3)]= -6

when m= -1 ß n= -1, zeroes are -1,(-1+3)=2, [-1+2(3)]=5

therefore, the value of m= 5 ß n= -3 or m= -1 ß n=3.

zeroes are -1,2,5.