if α and β are the zeroes of the polynomial 6x^2 - 7x -3, then form a quadratic polynomial whose zeroes are 2αand 2β
Answers
Answer:
Note: Let's consider a quadratic polynomial in variable x, say;
ax^2 + bx + c.
Now, if alpha and beta are the zeros of the polynomial ,then;
alpha+beta = -b/a and
alpha•beta = c/a
Here, the given quadratic polynomial is;
6x^2 - 7x - 3
Clearly, here we have;
a = 6
b = -7
c = -3
Thus,
alpha+beta = -b/a = -(-7/6) = 7/6 ,
alpha•beta = c/a = -3/6 = -1/2
Now, we need to obtain a quadratic equation whose zeros are ;
2•alpha and 2•beta.
Note: If A and B are the zeros of a quadratic polynomial, then the form of the polynomial is given as;
x^2 - (A+B)x + (A•B)
Here, it is given that, the zeros of the required polynomial are:
2•alpha and 2•beta.
Thus,
The required quadratic polynomial will be given as;
x^2-(2•alpha+2•beta)x+(2•alpha•2•beta)
=> x^2 - 2(alpha+beta)x + 4(alpha•beta)
=> x^2 - 2(7/6)x + 4(-1/2)
=> x^2 - (7/3)x - 2
=> x^2 - 7x/3 - 2
Thus, the required polynomial is;
x^2 - 7x/3 - 2
OR
(3x^2 - 7x - 6)/3
Answer:
I Hope this ans is useful
Step-by-step explanation:
Note: Let's consider a quadratic polynomial in variable x, say;
ax^2 + bx + c.
Now, if alpha and beta are the zeros of the polynomial ,then;
alpha+beta = -b/a and
alpha•beta = c/a
Here, the given quadratic polynomial is;
6x^2 - 7x - 3
Clearly, here we have;
a = 6
b = -7
c = -3
Thus,
alpha+beta = -b/a = -(-7/6) = 7/6 ,
alpha•beta = c/a = -3/6 = -1/2
Now, we need to obtain a quadratic equation whose zeros are ;
2•alpha and 2•beta.
Note: If A and B are the zeros of a quadratic polynomial, then the form of the polynomial is given as;
x^2 - (A+B)x + (A•B)
Here, it is given that, the zeros of the required polynomial are:
2•alpha and 2•beta.
Thus,
The required quadratic polynomial will be given as;
x^2-(2•alpha+2•beta)x+(2•alpha•2•beta)
=> x^2 - 2(alpha+beta)x + 4(alpha•beta)
=> x^2 - 2(7/6)x + 4(-1/2)
=> x^2 - (7/3)x - 2
=> x^2 - 7x/3 - 2
Thus, the required polynomial is;
x^2 - 7x/3 - 2
OR
(3x^2 - 7x - 6)/3