If the zero of the polynomial x2 + px + q and double of the zero of 2x2 - 5x-3 respectively find values of p and q?
if two polynomials have the same roots, one is a multiple of the other. So,
x^2+px+q = 2(x^2-(5/2)x-(3/2))
so, p = -5/2 and q = -3/2
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To find the values of p and q, let's start by finding the zeros of the given polynomials.
1. For the polynomial x^2 + px + q, let's assume the zeros are a and b. By Vieta's formulas, we know that:
a + b = -p ----(1)
a * b = q ----(2)
2. For the polynomial 2x^2 - 5x - 3, let's assume the zeros are c and d. We are given that one of the zeros is twice the other, so we can write:
c = 2d ----(3)
Now, let's find the values of a, b, c, and d using the information provided.
For the polynomial x^2 + px + q:
Since a and b are the zeros, the polynomial can be factored as (x - a)(x - b) = 0.
Expanding this equation:
x^2 - (a + b)x + ab = 0
Comparing it with the polynomial given: x^2 + px + q = 0
We get: -p = -(a + b) ----(4) (equating the coefficients of x)
q = ab ----(5) (equating the constant terms)
For the polynomial 2x^2 - 5x - 3:
Since c and d are the zeros, the polynomial can be factored as (x - c)(x - d) = 0.
Expanding this equation:
x^2 - (c + d)x + cd = 0
Comparing it with the polynomial given: 2x^2 - 5x - 3 = 0
We get: -5 = -(c + d) ----(6) (equating the coefficients of x)
-3 = cd ----(7) (equating the constant terms)
From equation (3), we have c = 2d. Substituting this into equation (6):
-5 = -((2d) + d)
-5 = -3d
d = 5/3
From equation (4), we have -p = -(a + b). Substituting the values of a = d = 5/3 and c = 2d into equation (1):
p = (5/3) + (2 * 5/3) = 15/3 + 10/3
p = 25/3
From equation (7), we have -3 = cd. Substituting the value of d = 5/3:
-3 = (5/3) * c
c = -3 * (3/5)
c = -9/5
Now, we have the values of p and q:
p = 25/3
q = ab
From equation (5), we have q = ab. Substituting the values of a = d = 5/3 and b = c = -9/5:
q = (5/3) * (-9/5)
q = -45/15
q = -3
Therefore, the values of p and q are:
p = 25/3
q = -3