13. Find the quadratic polynomial whose zeros are 2 and -6. Verify the
relation between the coefficients and the zeros of the polynomial.
Answers
Solution
Given :-
- Zeroes of any polynomial 2 & -6 .
Find :-
- Equation of polynomial ,
- relation between the coefficients and the zeros of the polynomial.
Explanation
Let, here
- first zeroes (p) = 2
- Second zeroes (q) = -6
So,
★ Sum of zeroes (p + q) = 2 + (-6)
==> Sum of zeroes (p + q) = -4.-----(1)
And,
★ Product of zeroes p . q = 2 * (-6)
==> Product of zeroes (p . q) = -12 -------(2)
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Equation ,formula
★ x² - ( sum of zeroes) x + product of zeroes = 0
Keep all above value,
==> x² - (-4)x + (-12) = 0
==> x² + 4x - 12 = 0
Hence, Required equation
- x² + 4x - 12 = 0
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Verification of zeroes & coefficient of polynomial
Important Formula
★ Sum of zeroes = -b/a
★ product of zeroes = c/a
Where,
- a = Coefficient of x²
- b = Coefficient of x
- c = Constant part
So,
==> Sum of zeroes = -4/1
==> p + q = -4 -------------(3)
And,
==> product of zeroes = -12/1
==> p . q = -12 --------------(4)
Here, equ(1) , equ(3) & equ(2) , equ(4) are same
Hence proved.
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Given that:
- Roots of a polynomial is 2 and -6.
To Find:
- The quadratic equation and verify the relationship between roots and coefficients.
Formula Used:
- For a quadratic equation f(x) having root as p and q, we have
f(x) = (x-p)(x-q) = 0
SOLUTION:
Let the quadratic equation be f(x).
So, using the above formula, we have
f(x) = (x-2)(x+6)
f(x) = x^2-2x+6x-12
f(x) = x^2+4x-12 = 0
So, the quadratic equation is x^2+4x-12 = 0.
Verification:
- Sum of roots = -b/a
- Product of roots = c/a
Now, ATQ, we have
b = 4 ,c = -12 and a = 1.
So, 2+(-6) = -4
-4 = -4...(1)
Also, (2)*(-6) = -12/1
-12 = -12...(2)
From eq(1) and eq(2), we get
LHS = RHS.
Hence, verified !