Let alpha and beta are the zeroes of quadratic polynomial 2x^2-5x-6 then form a quadratic polynomial whose zeroes are alpha + beta and alpha×beta?

Question

Answer

P(x)=2x²-5x-6,having zeros as alpha and beta,
to find -->f(x) such that its zeros are (alpha +beta)and alpha×beta,
in p(x) sum of zeros=alpha+beta=-b/a,
alpha+beta=-(-5)/2=5/2,
product of zeros=alpha×beta=c/a,
alpha×beta=-6/2=-3,
zeros of f(x) are 5/2 and -3,
f(x)=x²-( sum of zeros)x+ product of zeros,
=x²-(5/2-3)x+(5/2×-3),
=x²-(-1/2)x-(15/2),
=x²+1/2x-15/2,
=2x²+x-15

Answer

Answer

Since alpha and beta are the zeroes of the polynomial 2x^2 -5x -6 therefore alpha + beta = -b/a = -(-5)/2 = 5/2and alpha x beta = c/a = -6/2 = -3
since alpha + beta and alpha x beta are zeroes of another polynomial
therefore sum of zeroes = 5/2 + (-3) = -1/2product of zeroes = 5/2 x (-3) = -15/2
therefore the polynomial would be:
x^2 - (sum of zeroes)x + product of zeroes= x^2 + 1/2x -15/2= 2x^2 + x - 15
therefore the required polynomial is 2x^2 + x -15

Answer

How to Form a Quadratic Polynomial with Given Zeroes

Given Information:

Quadratic polynomial: 2x^2-5x-6

Zeroes: alpha and beta

Finding the Value of Alpha and Beta

Using the quadratic formula, we can find the values of alpha and beta:

x = (-b ± √(b^2-4ac))/2a

For the given quadratic polynomial, a = 2, b = -5, and c = -6

Substituting these values in the quadratic formula, we get:

alpha = (-(-5) + √((-5)^2-4(2)(-6)))/2(2) = 3

beta = (-(-5) - √((-5)^2-4(2)(-6)))/2(2) = -1.5

Finding the Value of Alpha×Beta

Using the product of roots formula, we can find the value of alpha×beta:

alpha×beta = c/a = -6/2 = -3

Forming the Quadratic Polynomial

Let the quadratic polynomial with zeroes alpha, beta, and alpha×beta be ax^2 + bx + c

We know that:

alpha + beta = b/a

alpha×beta = c/a

Substituting the values of alpha, beta, and alpha×beta, we get:

3 + (-1.5) + (-3) = b/2

b = -1

-3 = c/2

c = -6

Therefore, the quadratic polynomial with zeroes alpha, beta, and alpha×beta is:

2x^2 - x - 6

Final Answer:

The quadratic polynomial with zeroes alpha, beta, and alpha×beta is 2x^2 - x - 6.

Answer

The zeroes of the polynomial x2-3x-m(m+3) are

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